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Partial fractions practice problems

Split a rational function into simple pieces, then integrate each. 59 problems with worked solutions; in 31 of them every equation is proved by a computer algebra system.

19 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 19 of these also have a second solution, written separately and checked the same way. How we verify

Problem 4.220

hard Video
Find \( \displaystyle \int \frac{3 x + 6}{3 x + \left(3 x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{3 x + 6}{3 x + \left(3 x + 1\right)^{2} - 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{3 x + 6}{9 x^{2} + 9 x}\, dx \]
    algebra algebraExpand the squared term in the denominator. Simplify the denominator by combining like terms.✓ Proved
  3. \[ = \int \frac{3 x + 6}{9 x \left(x + 1\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \frac{x + 2}{3 x \left(x + 1\right)}\, dx \]
    simplifyDivide the numerator and denominator by 3.✓ Proved
  5. \[ = \int \left(- \frac{1}{3 x + 3} + \frac{2}{3 x}\right)\, dx \]
    partial-fractionsPerform partial fraction decomposition.✓ Proved
  6. \[ = \int \frac{2}{3 x}\, dx - \int \frac{1}{3 x + 3}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  7. \[ = \frac{2 \ln{\left(x \right)}}{3} - \frac{\ln{\left(x + 1 \right)}}{3} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)}}{3} + C \)
Top: the graph of F = 2log(x)/3 - log(x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x + 6)/(3x + (3x + 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{3 x + 6}{3 x + \left(3 x + 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{x + 2}{3 x \left(x + 1\right)}\, dx \]
    simplifySimplify numerator and denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{3 x + 3} + \frac{2}{3 x}\right)\, dx \]
    partial-fractionsDecompose into partial fractions.✓ Proved
  4. \[ = \frac{2 \int \frac{1}{x}\, dx}{3} - \frac{\int \frac{1}{x + 1}\, dx}{3} \]
    linearitySplit the integral.✓ Proved
  5. \[ = \frac{2 \ln{\left(x \right)}}{3} - \frac{\ln{\left(x + 1 \right)}}{3} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.276

medium Video
Find \( \displaystyle \int \frac{2 x - 1}{2 x \left(2 x + 2\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{2 x - 1}{2 x \left(2 x + 2\right)}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{2 x - 1}{4 x^{2} + 4 x}\, dx \]
    algebraExpand the denominator.✓ Proved
  3. \[ = \int \left(\frac{2 x}{4 x^{2} + 4 x} - \frac{1}{4 x^{2} + 4 x}\right)\, dx \]
    linearitySplit the integrand into two parts.✓ Proved
  4. \[ = \int \left(- \frac{1}{4 x^{2} + 4 x} + \frac{1}{2 x + 2}\right)\, dx \]
    simplifySimplify the first term by canceling x.✓ Proved
  5. \[ = \int \frac{1}{2 x + 2}\, dx - \int \frac{1}{4 x^{2} + 4 x}\, dx \]
    linearitySplit the integral again.✓ Proved
  6. \[ = - \int \frac{1}{4 x \left(x + 1\right)}\, dx + \int \frac{1}{2 x + 2}\, dx \]
    algebraFactor out the constant from the second denominator.✓ Proved
  7. \[ = \int \frac{1}{2 x + 2}\, dx - \int \frac{1}{4 x^{2} + 4 x}\, dx \]
    algebraFactor out 4 from the second term.✓ Proved
  8. \[ = - \frac{\int \frac{1}{x \left(x + 1\right)}\, dx}{4} + \int \frac{1}{2 x + 2}\, dx \]
    linearityPull out the constant factor.✓ Proved
  9. \[ = \int \frac{1}{2 x + 2}\, dx - \frac{\int \left(- \frac{1}{x + 1} + \frac{1}{x}\right)\, dx}{4} \]
    partial-fractionsApply partial fraction decomposition to the second term.✓ Proved
  10. \[ = - \frac{\int \frac{1}{x}\, dx}{4} + \frac{\int \frac{1}{x + 1}\, dx}{4} + \int \frac{1}{2 x + 2}\, dx \]
    linearityDistribute the constant and split the integral.✓ Proved
  11. \[ = - \frac{\ln{\left(x \right)}}{4} + \frac{\ln{\left(x + 1 \right)}}{4} + \frac{\ln{\left(2 x + 2 \right)}}{2} \]
    antiderivativeEvaluate the integrals using the natural logarithm rule.✓ Proved
Answer \( \frac{- \ln{\left(x \right)} + 3 \ln{\left(x + 1 \right)}}{4} + C \)
Top: the graph of F = -log(x)/4 + 3log(x + 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)/(2x·(2x + 2)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{2 x - 1}{2 x \left(2 x + 2\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{2 x - 1}{4 x \left(x + 1\right)}\, dx \]
    simplifyCombine constants in the denominator.✓ Proved
  3. \[ = \int \left(\frac{3}{4 x + 4} - \frac{1}{4 x}\right)\, dx \]
    partial-fractionsDecompose into simpler fractions.✓ Proved
  4. \[ = - \frac{\ln{\left(x \right)}}{4} + \frac{3 \ln{\left(x + 1 \right)}}{4} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1476

medium Video
Find \( \displaystyle \int \frac{x + 6}{x + \left(x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{x + 6}{x + \left(x + 1\right)^{2} - 1}\, dx \]
    simplifyStart with the given integral.✓ Proved
  2. \[ = \int \frac{x + 6}{x^{2} + 3 x}\, dx \]
    algebra algebraExpand the squared term. Combine like terms in the denominator.✓ Proved
  3. \[ = \int \frac{x + 6}{x \left(x + 3\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \left(\frac{1}{x + 3} + \frac{6}{x \left(x + 3\right)}\right)\, dx \]
    linearity simplifySplit the integrand into two parts. Simplify the second term by canceling x.✓ Proved
  5. \[ = \int \frac{6}{x \left(x + 3\right)}\, dx + \int \frac{1}{x + 3}\, dx \]
    linearitySplit the integral again.✓ Proved
  6. \[ = \int \frac{1}{x + 3}\, dx + \int \left(- \frac{2}{x + 3} + \frac{2}{x}\right)\, dx \]
    partial-fractionsApply partial fraction decomposition to the first term.✓ Proved
  7. \[ = \int \frac{2}{x}\, dx + \int \frac{1}{x + 3}\, dx - \int \frac{2}{x + 3}\, dx \]
    linearityDistribute the integral sign.✓ Proved
  8. \[ = \int \frac{2}{x}\, dx - \int \frac{1}{x + 3}\, dx \]
    algebraCombine the integrals of the (x+3) terms.✓ Proved
  9. \[ = 2 \ln{\left(x \right)} - \ln{\left(x + 3 \right)} \]
    antiderivativeEvaluate the remaining integrals.✓ Proved
Answer \( 2 \ln{\left(x \right)} - \ln{\left(x + 3 \right)} + C \)
Top: the graph of F = 2log(x) - log(x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 6)/(x + (x + 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{x + 6}{x + \left(x + 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{x + 6}{x^{2} + 3 x}\, dx \]
    algebraSimplify the denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{x + 3} + \frac{2}{x}\right)\, dx \]
    partial-fractionsDecompose into partial fractions.✓ Proved
  4. \[ = 2 \int \frac{1}{x}\, dx - \int \frac{1}{x + 3}\, dx \]
    linearitySplit the integral.✓ Proved
  5. \[ = 2 \ln{\left(x \right)} - \ln{\left(x + 3 \right)} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.227

medium Video
Find \( \displaystyle \int \frac{\left(x - 1\right)^{2} + 1}{\left(x - 2\right) \left(x - 1\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{\left(x - 1\right)^{2} + 1}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{x^{2} - 2 x + 2}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    algebraExpand the numerator.✓ Proved
  3. \[ = \int \frac{x + \left(x - 2\right) \left(x - 1\right)}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    algebraRewrite the numerator to facilitate division.✓ Proved
  4. \[ = \int \left(\frac{x}{\left(x - 2\right) \left(x - 1\right)} + 1\right)\, dx \]
    long-division algebraPerform polynomial division. Split the numerator of the remainder term.✓ Proved
  5. \[ = \int \left(1 + \frac{1}{x - 1} + \frac{2}{\left(x - 2\right) \left(x - 1\right)}\right)\, dx \]
    algebraSimplify the fraction by canceling (x - 2).✓ Proved
  6. \[ = \int \left(1 - \frac{1}{x - 1} + \frac{2}{x - 2}\right)\, dx \]
    partial-fractions simplifyApply partial fraction decomposition to the last term. Combine like terms.✓ Proved
  7. \[ = \int 1\, dx + \int \frac{2}{x - 2}\, dx - \int \frac{1}{x - 1}\, dx \]
    linearityDistribute the integral across the terms.✓ Proved
  8. \[ = x + 2 \ln{\left(x - 2 \right)} - \ln{\left(x - 1 \right)} \]
    antiderivativeEvaluate the integrals.✓ Proved
Answer \( x + 2 \ln{\left(x - 2 \right)} - \ln{\left(x - 1 \right)} + C \)
Top: the graph of F = x + 2log(x - 2) - log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = ((x - 1)² + 1)/((x - 2)·(x - 1)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{\left(x - 1\right)^{2} + 1}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{x^{2} - 2 x + 2}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    algebra✓ Proved
  3. \[ = \int \frac{x^{2} - 2 x + 2}{x^{2} - 3 x + 2}\, dx \]
    algebra✓ Proved
  4. \[ = x + \int \frac{x}{x^{2} - 3 x + 2}\, dx \]
    long-divisionIntegral of 1 is x✓ Proved
  5. \[ = x + \int \frac{x}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    rewrite✓ Proved
  6. \[ = x + \int \left(- \frac{1}{x - 1} + \frac{2}{x - 2}\right)\, dx \]
    partial-fractions✓ Proved
  7. \[ = x + \int \frac{2}{x - 2}\, dx + \int \left(- \frac{1}{x - 1}\right)\, dx \]
    linearity✓ Proved
  8. \[ = x + 2 \ln{\left(x - 2 \right)} - \ln{\left(x - 1 \right)} \]
    antiderivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.264

medium Video
Find \( \displaystyle \int \frac{4 x + 3}{\left(2 x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{4 x + 3}{\left(2 x + 1\right)^{2} - 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{4 x + 3}{4 x^{2} + 4 x}\, dx \]
    algebraExpand the denominator.✓ Proved
  3. \[ = \int \frac{4 x + 3}{4 x \left(x + 1\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \left(\frac{1}{x + 1} + \frac{3}{4 x \left(x + 1\right)}\right)\, dx \]
    algebra simplifySplit the numerator. Simplify the first term.✓ Proved
  5. \[ = \int \left(\frac{1}{4 \left(x + 1\right)} + \frac{3}{4 x}\right)\, dx \]
    partial-fractionsApply partial fraction decomposition to the second term.✓ Proved
  6. \[ = \int \left(- \frac{3}{4 x + 4} + \frac{1}{x + 1} + \frac{3}{4 x}\right)\, dx \]
    algebraDistribute the constant.✓ Proved
  7. \[ = \int \left(\frac{1}{4 x + 4} + \frac{3}{4 x}\right)\, dx \]
    simplifyCombine the terms with (x + 1) in the denominator.✓ Proved
  8. \[ = \int \frac{3}{4 x}\, dx + \int \frac{1}{4 x + 4}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  9. \[ = \frac{3 \ln{\left(x \right)}}{4} + \frac{\ln{\left(x + 1 \right)}}{4} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{3 \ln{\left(x \right)} + \ln{\left(x + 1 \right)}}{4} + C \)
Top: the graph of F = 3log(x)/4 + log(x + 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (4x + 3)/((2x + 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{4 x + 3}{\left(2 x + 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{4 x + 3}{4 x \left(x + 1\right)}\, dx \]
    rewriteRewrite the denominator as 4*x*(x+1).✓ Proved
  3. \[ = \int \left(\frac{1}{4 \left(x + 1\right)} + \frac{3}{4 x}\right)\, dx \]
    partial-fractionsDecompose into partial fractions.✓ Proved
  4. \[ = \frac{3 \ln{\left(x \right)}}{4} + \frac{\ln{\left(x + 1 \right)}}{4} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.644

medium Video
Find \( \displaystyle \int \frac{3 x + 2}{- x + \left(x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{3 x + 2}{- x + \left(x + 1\right)^{2} - 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{3 x + 2}{x^{2} + x}\, dx \]
    algebra simplifyExpand the squared term. Simplify the denominator.✓ Proved
  3. \[ = \int \frac{3 x + 2}{x \left(x + 1\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \left(\frac{3}{x + 1} + \frac{2}{x \left(x + 1\right)}\right)\, dx \]
    linearity simplifySplit the numerator. Cancel x from the first term.✓ Proved
  5. \[ = \int \left(\frac{1}{x + 1} + \frac{2}{x}\right)\, dx \]
    partial-fractions simplifyApply partial fraction decomposition to the second term. Combine the terms involving 1/(x + 1).✓ Proved
  6. \[ = 2 \int \frac{1}{x}\, dx + \int \frac{1}{x + 1}\, dx \]
    linearitySplit the integral.✓ Proved
  7. \[ = 2 \ln{\left(x \right)} + \ln{\left(x + 1 \right)} \]
    antiderivativeIntegrate each term.✓ Proved
Answer \( 2 \ln{\left(x \right)} + \ln{\left(x + 1 \right)} + C \)
Top: the graph of F = 2log(x) + log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x + 2)/(-x + (x + 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{3 x + 2}{- x + \left(x + 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{3 x + 2}{x^{2} + x}\, dx \]
    rewriteSimplify the denominator.✓ Proved
  3. \[ = \int \frac{3 x + 2}{x \left(x + 1\right)}\, dx \]
    algebraFactor the quadratic x**2 + x.✓ Proved
  4. \[ = \int \left(\frac{1}{x + 1} + \frac{2}{x}\right)\, dx \]
    partial-fractionsDecompose into simpler fractions.✓ Proved
  5. \[ = \int \frac{2}{x}\, dx + \int \frac{1}{x + 1}\, dx \]
    linearitySplit the integral.✓ Proved
  6. \[ = 2 \int \frac{1}{x}\, dx + \int \frac{1}{x + 1}\, dx \]
    algebraTake the constant 2 outside.✓ Proved
  7. \[ = 2 \ln{\left(x \right)} + \ln{\left(x + 1 \right)} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.229

medium Video
Find \( \displaystyle \int \frac{\left(x + 1\right)^{2} + 1}{x \left(x + 1\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \left(\frac{x + 1}{x} + \frac{1}{x \left(x + 1\right)}\right)\, dx \]
    algebra simplifySplit the fraction into two parts. Simplify the first term.✓ Proved
  2. \[ = \int \left(1 + \frac{1}{x} + \frac{1}{x \left(x + 1\right)}\right)\, dx \]
    algebraSplit the first term further.✓ Proved
  3. \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \frac{1}{x \left(x + 1\right)}\, dx \]
    linearityApply linearity to the integral.✓ Proved
  4. \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \left(- \frac{1}{x + 1} + \frac{1}{x}\right)\, dx \]
    partial-fractionsUse partial fraction decomposition on the third term.✓ Proved
  5. \[ = \int 1\, dx + 2 \int \frac{1}{x}\, dx - \int \frac{1}{x + 1}\, dx \]
    linearity simplifyDistribute the integral. Combine the like terms.✓ Proved
  6. \[ = x + 2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)} \]
    antiderivativeIntegrate each term.✓ Proved
Answer \( x + 2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)} + C \)
Top: the graph of F = x + 2log(x) - log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = ((x + 1)² + 1)/(x·(x + 1)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{\left(x + 1\right)^{2} + 1}{x \left(x + 1\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{x^{2} + 2 x + 2}{x \left(x + 1\right)}\, dx \]
    rewriteExpand the numerator.✓ Proved
  3. \[ = \int \left(1 + \frac{x + 2}{x \left(x + 1\right)}\right)\, dx \]
    long-divisionDivide the rational function.✓ Proved
  4. \[ = \int \left(1 - \frac{1}{x + 1} + \frac{2}{x}\right)\, dx \]
    partial-fractionsDecompose the remaining fraction.✓ Proved
  5. \[ = \int 1\, dx + 2 \int \frac{1}{x}\, dx - \int \frac{1}{x + 1}\, dx \]
    linearitySplit the integral.✓ Proved
  6. \[ = x + 2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1402

medium Video
Find \( \displaystyle \int \frac{x}{\left(x + 1\right) \left(x + 3\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{x}{\left(x + 1\right) \left(x + 3\right)}\, dx \]
    integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate decomposition.✓ Proved
  2. \[ = \int \left(\frac{1}{x + 3} - \frac{1}{\left(x + 1\right) \left(x + 3\right)}\right)\, dx \]
    linearity simplifySplit the integral into two parts. Simplify the first term by canceling (x + 1).✓ Proved
  3. \[ = \int \left(\frac{1}{2 x + 6} - \frac{1}{2 x + 2} + \frac{1}{x + 3}\right)\, dx \]
    partial-fractions algebraApply partial fraction decomposition to the second term. Distribute the negative sign.✓ Proved
  4. \[ = \int \left(\frac{3}{2 x + 6} - \frac{1}{2 x + 2}\right)\, dx \]
    algebraCombine the terms involving (x + 3).✓ Proved
  5. \[ = - \frac{\ln{\left(x + 1 \right)}}{2} + \frac{3 \ln{\left(x + 3 \right)}}{2} \]
    antiderivativeThe antiderivative of 1/(x+a) is log(x+a).✓ Proved
Answer \( \frac{- \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 3 \right)}}{2} + C \)
Top: the graph of F = -log(x + 1)/2 + 3log(x + 3)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = x/((x + 1)·(x + 3)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{x}{\left(x + 1\right) \left(x + 3\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \left(\frac{3}{2 \left(x + 3\right)} - \frac{1}{2 \left(x + 1\right)}\right)\, dx \]
    partial-fractionsRewrite the integrand as a sum of simple fractions✓ Proved
  3. \[ = - \frac{\int \frac{1}{x + 1}\, dx}{2} + \frac{3 \int \frac{1}{x + 3}\, dx}{2} \]
    linearityApply linearity of the integral✓ Proved
  4. \[ = - \frac{\ln{\left(x + 1 \right)}}{2} + \frac{3 \ln{\left(x + 3 \right)}}{2} \]
    antiderivativeIntegrate each term✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1244

medium Video
Find \( \displaystyle \int \frac{3 x - 4}{- x + \left(x - 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{3 x - 4}{- x + \left(x - 1\right)^{2} + 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{3 x - 4}{x^{2} - 3 x + 2}\, dx \]
    algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
  3. \[ = \int \frac{3 x - 4}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    algebraFactor the quadratic denominator.✓ Proved
  4. \[ = \int \left(\frac{1}{x - 1} + \frac{2}{x - 2}\right)\, dx \]
    partial-fractionsApply partial fraction decomposition.✓ Proved
  5. \[ = \int \frac{2}{x - 2}\, dx + \int \frac{1}{x - 1}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  6. \[ = 2 \ln{\left(x - 2 \right)} + \ln{\left(x - 1 \right)} \]
    antiderivativeIntegrate each term.✓ Proved
Answer \( 2 \ln{\left(x - 2 \right)} + \ln{\left(x - 1 \right)} + C \)
Top: the graph of F = 2log(x - 2) + log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x - 4)/(-x + (x - 1)² + 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{3 x - 4}{- x + \left(x - 1\right)^{2} + 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{3 x - 4}{x^{2} - 3 x + 2}\, dx \]
    rewrite✓ Proved
  3. \[ = \int \frac{3 x - 4}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
    algebraFactor the quadratic denominator✓ Proved
  4. \[ = \int \left(\frac{1}{x - 1} + \frac{2}{x - 2}\right)\, dx \]
    partial-fractions✓ Proved
  5. \[ = \int \frac{2}{x - 2}\, dx + \int \frac{1}{x - 1}\, dx \]
    linearity✓ Proved
  6. \[ = \ln{\left(x - 1 \right)} + \int \frac{2}{x - 2}\, dx \]
    antiderivative✓ Proved
  7. \[ = 2 \ln{\left(x - 2 \right)} + \ln{\left(x - 1 \right)} \]
    antiderivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.216

medium Video
Find \( \displaystyle \int \frac{2 x - 1}{\left(x - 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{2 x - 1}{\left(x - 1\right)^{2} - 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{2 x - 1}{x^{2} - 2 x}\, dx \]
    algebraExpand the denominator.✓ Proved
  3. \[ = \int \frac{2 x - 1}{x \left(x - 2\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \left(\frac{3}{2 \left(x - 2\right)} + \frac{1}{2 x}\right)\, dx \]
    partial-fractionsPerform partial fraction decomposition.✓ Proved
  5. \[ = \int \frac{1}{2 x}\, dx + \int \frac{3}{2 \left(x - 2\right)}\, dx \]
    linearitySplit the integral using linearity.✓ Proved
  6. \[ = \frac{\ln{\left(x \right)}}{2} + \frac{3 \ln{\left(x - 2 \right)}}{2} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x \right)} + 3 \ln{\left(x - 2 \right)}}{2} + C \)
Top: the graph of F = log(x)/2 + 3log(x - 2)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)/((x - 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{2 x - 1}{\left(x - 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{2 x - 1}{x \left(x - 2\right)}\, dx \]
    rewriteSimplify the denominator.✓ Proved
  3. \[ = \int \left(\frac{3}{2 x - 4} + \frac{1}{2 x}\right)\, dx \]
    partial-fractionsDecompose into simpler fractions.✓ Proved
  4. \[ = \frac{\int \frac{1}{x}\, dx}{2} + \frac{3 \int \frac{1}{x - 2}\, dx}{2} \]
    linearitySeparate the integrals.✓ Proved
  5. \[ = \frac{\ln{\left(x \right)}}{2} + \frac{3 \ln{\left(x - 2 \right)}}{2} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.364

medium Video
Find \( \displaystyle \int \frac{1}{9 x^{2} + 9 x} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{9 x^{2} + 9 x}\, dx \]
    integral algebraStart with the integral of the given function. Factor out the constant 9.✓ Proved
  2. \[ = \int \frac{1}{9 x \left(x + 1\right)}\, dx \]
    algebraFactor the quadratic expression.✓ Proved
  3. \[ = \int \left(- \frac{1}{9 x + 9} + \frac{1}{9 x}\right)\, dx \]
    partial-fractionsDecompose the integrand using partial fractions.✓ Proved
  4. \[ = \int \frac{1}{9 x}\, dx - \int \frac{1}{9 x + 9}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  5. \[ = \frac{\ln{\left(x \right)}}{9} - \frac{\ln{\left(x + 1 \right)}}{9} \]
    antiderivativeIntegrate each term using the natural logarithm rule.✓ Proved
Answer \( \frac{\ln{\left(x \right)} - \ln{\left(x + 1 \right)}}{9} + C \)
Top: the graph of F = log(x)/9 - log(x + 1)/9. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/(9x² + 9x).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{9 x^{2} + 9 x}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{1}{9 x \left(x + 1\right)}\, dx \]
    rewriteFactor the denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{9 \left(x + 1\right)} + \frac{1}{9 x}\right)\, dx \]
    partial-fractionsDecompose 1/(x(x+1)).✓ Proved
  4. \[ = \frac{\int \frac{1}{x}\, dx}{9} - \frac{\int \frac{1}{x + 1}\, dx}{9} \]
    linearitySplit the integral.✓ Proved
  5. \[ = \frac{\ln{\left(x \right)}}{9} - \frac{\ln{\left(x + 1 \right)}}{9} \]
    antiderivative simplifyIntegrate 1/x and 1/(x+1). Combine the terms.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.646

medium Video
Find \( \displaystyle \int \frac{1}{4 x^{2} - 4} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{4 x^{2} - 4}\, dx \]
    integral algebraStart with the integral of the given function. Factor out the constant 4 from the denominator.✓ Proved
  2. \[ = \int \frac{1}{\left(x + 1\right) \left(4 x - 4\right)}\, dx \]
    algebraFactor the difference of squares.✓ Proved
  3. \[ = \int \left(- \frac{1}{8 x + 8} + \frac{1}{8 x - 8}\right)\, dx \]
    partial-fractionsDecompose the integrand using partial fraction decomposition.✓ Proved
  4. \[ = \int \frac{1}{8 x - 8}\, dx - \int \frac{1}{8 x + 8}\, dx \]
    linearitySplit the integral into two separate terms.✓ Proved
  5. \[ = \frac{\ln{\left(x - 1 \right)}}{8} - \frac{\ln{\left(x + 1 \right)}}{8} \]
    antiderivativeIntegrate each term using the rule for 1/u.✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)} - \ln{\left(x + 1 \right)}}{8} + C \)
Top: the graph of F = log(x - 1)/8 - log(x + 1)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/(4x² - 4).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{4 x^{2} - 4}\, dx \]
    integral algebra✓ Proved
  2. \[ = \int \frac{1}{4 \left(x^{2} - 1\right)}\, dx \]
    algebra✓ Proved
  3. \[ = \frac{\int \frac{1}{x^{2} - 1}\, dx}{4} \]
    linearity✓ Proved
  4. \[ = \frac{\int \left(- \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}\right)\, dx}{4} \]
    partial-fractions✓ Proved
  5. \[ = \frac{\int \left(- \frac{1}{x + 1} + \frac{1}{x - 1}\right)\, dx}{8} \]
    linearity✓ Proved
  6. \[ = \frac{\int \frac{1}{x - 1}\, dx}{8} - \frac{\int \frac{1}{x + 1}\, dx}{8} \]
    linearity✓ Proved
  7. \[ = \frac{\ln{\left(x - 1 \right)}}{8} - \frac{\ln{\left(x + 1 \right)}}{8} \]
    antiderivative simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.739

medium Video
Find \( \displaystyle \int \frac{4 x - 1}{\left(2 x - 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{4 x - 1}{\left(2 x - 1\right)^{2} - 1}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{4 x - 1}{4 x^{2} - 4 x}\, dx \]
    algebraExpand the denominator.✓ Proved
  3. \[ = \int \frac{4 x - 1}{4 x \left(x - 1\right)}\, dx \]
    simplifyFactor the denominator.✓ Proved
  4. \[ = \int \frac{1}{4 x}\, dx + \int \frac{3}{4 x - 4}\, dx \]
    partial-fractionsPerform partial fraction decomposition.✓ Proved
  5. \[ = \int \frac{1}{4 x}\, dx + \frac{3 \int \frac{1}{x - 1}\, dx}{4} \]
    linearityApply linearity to the integral.✓ Proved
  6. \[ = \frac{\ln{\left(x \right)}}{4} + \frac{3 \ln{\left(x - 1 \right)}}{4} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x \right)} + 3 \ln{\left(x - 1 \right)}}{4} + C \)
Top: the graph of F = log(x)/4 + 3log(x - 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (4x - 1)/((2x - 1)² - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{4 x - 1}{\left(2 x - 1\right)^{2} - 1}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{4 x - 1}{4 x \left(x - 1\right)}\, dx \]
    rewriteRewrite the denominator (2*x - 1)**2 - 1 as 4*x*(x - 1).✓ Proved
  3. \[ = \int \left(\frac{3}{4 x - 4} + \frac{1}{4 x}\right)\, dx \]
    partial-fractionsDecompose into partial fractions.✓ Proved
  4. \[ = \int \frac{1}{4 x}\, dx + \int \frac{3}{4 x - 4}\, dx \]
    linearitySplit the integral.✓ Proved
  5. \[ = \frac{\int \frac{1}{x}\, dx}{4} + \frac{3 \int \frac{1}{x - 1}\, dx}{4} \]
    linearityFactor constants out of each integral.✓ Proved
  6. \[ = \frac{\ln{\left(x \right)}}{4} + \frac{3 \ln{\left(x - 1 \right)}}{4} \]
    antiderivativeIntegrate 1/x and 1/(x-1).✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.215

medium Video
Find \( \displaystyle \int \frac{3 x - 1}{x^{2} - x} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{3 x - 1}{x^{2} - x}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{3 x - 1}{x \left(x - 1\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  3. \[ = \int \left(\frac{2}{x - 1} + \frac{1}{x}\right)\, dx \]
    partial-fractionsDecompose the integrand using partial fractions.✓ Proved
  4. \[ = \int \frac{1}{x}\, dx + \int \frac{2}{x - 1}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  5. \[ = \ln{\left(x \right)} + 2 \ln{\left(x - 1 \right)} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \ln{\left(x \right)} + 2 \ln{\left(x - 1 \right)} + C \)
Top: the graph of F = log(x) + 2log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x - 1)/(x² - x).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{3 x - 1}{x^{2} - x}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{3 x - 1}{x \left(x - 1\right)}\, dx \]
    algebraRewrite the denominator as a product✓ Proved
  3. \[ = \int \left(\frac{2}{x - 1} + \frac{1}{x}\right)\, dx \]
    partial-fractionsDecompose into simpler fractions✓ Proved
  4. \[ = \int \frac{1}{x}\, dx + \int \frac{2}{x - 1}\, dx \]
    linearitySeparate the integrals✓ Proved
  5. \[ = \ln{\left(x \right)} + 2 \ln{\left(x - 1 \right)} \]
    antiderivativeIntegrate each term✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.268

medium Video
Find \( \displaystyle \int \frac{1}{3 x \left(3 x + 3\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{3 x \left(3 x + 3\right)}\, dx \]
    integralStart with the given integral.✓ Proved
  2. \[ = \int \frac{1}{9 x \left(x + 1\right)}\, dx \]
    algebraFactor out 3 from the second term in the denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{9 x + 9} + \frac{1}{9 x}\right)\, dx \]
    partial-fractionsUse partial fraction decomposition.✓ Proved
  4. \[ = \int \frac{1}{9 x}\, dx - \int \frac{1}{9 x + 9}\, dx \]
    linearitySplit the integral into two parts.✓ Proved
  5. \[ = \frac{\ln{\left(x \right)}}{9} - \frac{\ln{\left(x + 1 \right)}}{9} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x \right)} - \ln{\left(x + 1 \right)}}{9} + C \)
Top: the graph of F = log(x)/9 - log(x + 1)/9. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/(3x·(3x + 3)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{3 x \left(3 x + 3\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{1}{9 x \left(x + 1\right)}\, dx \]
    rewriteFactor 3*x+3 as 3*(x+1) and simplify the denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{9 \left(x + 1\right)} + \frac{1}{9 x}\right)\, dx \]
    partial-fractionsDecompose 1/(x*(x+1)) into 1/x - 1/(x+1).✓ Proved
  4. \[ = \frac{\int \frac{1}{x}\, dx}{9} - \frac{\int \frac{1}{x + 1}\, dx}{9} \]
    linearitySplit the integral and factor constants.✓ Proved
  5. \[ = \frac{\ln{\left(x \right)}}{9} - \frac{\ln{\left(x + 1 \right)}}{9} \]
    antiderivativeIntegrate 1/x and 1/(x+1).✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.803

medium Video
Find \( \displaystyle \int \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \frac{2 x + 1}{4 x^{2} + 12 x + 8}\, dx \]
    algebraExpand the denominator.✓ Proved
  3. \[ = \int \frac{2 x + 1}{\left(x + 2\right) \left(4 x + 4\right)}\, dx \]
    algebraFactor the denominator.✓ Proved
  4. \[ = \int \left(\frac{3}{4 x + 8} - \frac{1}{4 x + 4}\right)\, dx \]
    partial-fractionsDecompose the integrand using partial fractions.✓ Proved
  5. \[ = - \frac{\ln{\left(x + 1 \right)}}{4} + \frac{3 \ln{\left(x + 2 \right)}}{4} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{- \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 2 \right)}}{4} + C \)
Top: the graph of F = -log(x + 1)/4 + 3log(x + 2)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)/((2x + 2)·(2x + 4)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \left(\frac{3}{4 x + 8} - \frac{1}{4 x + 4}\right)\, dx \]
    partial-fractions✓ Proved
  3. \[ = \int \left(- \frac{1}{4 x + 4}\right)\, dx + \int \frac{3}{4 x + 8}\, dx \]
    linearity✓ Proved
  4. \[ = - \frac{\int \frac{1}{x + 1}\, dx}{4} + \frac{3 \int \frac{1}{x + 2}\, dx}{4} \]
    linearity✓ Proved
  5. \[ = - \frac{\ln{\left(x + 1 \right)}}{4} + \frac{3 \ln{\left(x + 2 \right)}}{4} \]
    antiderivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.881

medium Video
Find \( \displaystyle \int \frac{1}{\left(x + 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{\left(x + 1\right)^{2} - 4}\, dx \]
    integralStart with the given integral.✓ Proved
  2. \[ = \int \frac{1}{x^{2} + 2 x - 3}\, dx \]
    algebraExpand the squared term.✓ Proved
  3. \[ = \int \frac{1}{\left(x - 1\right) \left(x + 3\right)}\, dx \]
    algebraFactor the quadratic expression.✓ Proved
  4. \[ = \int \left(- \frac{1}{4 \left(x + 3\right)} + \frac{1}{4 \left(x - 1\right)}\right)\, dx \]
    partial-fractionsPerform partial fraction decomposition.✓ Proved
  5. \[ = \frac{\ln{\left(x - 1 \right)}}{4} - \frac{\ln{\left(x + 3 \right)}}{4} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x - 1 \right)} - \ln{\left(x + 3 \right)}}{4} + C \)
Top: the graph of F = log(x - 1)/4 - log(x + 3)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/((x + 1)² - 4).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{\left(x + 1\right)^{2} - 4}\, dx \]
    integral✓ Proved
  2. \[ = \int \frac{1}{\left(x - 1\right) \left(x + 3\right)}\, dx \]
    rewriteFactor the quadratic denominator.✓ Proved
  3. \[ = \int \left(- \frac{1}{4 \left(x + 3\right)} + \frac{1}{4 \left(x - 1\right)}\right)\, dx \]
    partial-fractionsDecompose into partial fractions.✓ Proved
  4. \[ = \frac{\int \frac{1}{x - 1}\, dx}{4} - \frac{\int \frac{1}{x + 3}\, dx}{4} \]
    linearitySplit the integral.✓ Proved
  5. \[ = \frac{\ln{\left(x - 1 \right)}}{4} - \frac{\ln{\left(x + 3 \right)}}{4} \]
    antiderivativeIntegrate each term.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1403

medium Video
Find \( \displaystyle \int \frac{1}{\left(x - 2\right) \left(x + 1\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{\left(x - 2\right) \left(x + 1\right)}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \left(- \frac{1}{3 \left(x + 1\right)} + \frac{1}{3 \left(x - 2\right)}\right)\, dx \]
    partial-fractionsPerform partial fraction decomposition.✓ Proved
  3. \[ = \int \left(- \frac{1}{3 x + 3} + \frac{1}{3 x - 6}\right)\, dx \]
    algebraSimplify the expression by factoring out common constants.✓ Proved
  4. \[ = \int \frac{1}{3 x - 6}\, dx - \int \frac{1}{3 x + 3}\, dx \]
    linearitySplit the integral into two parts using linearity.✓ Proved
  5. \[ = \frac{\ln{\left(x - 2 \right)}}{3} - \frac{\ln{\left(x + 1 \right)}}{3} \]
    antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{\ln{\left(x - 2 \right)} - \ln{\left(x + 1 \right)}}{3} + C \)
Top: the graph of F = log(x - 2)/3 - log(x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/((x - 2)·(x + 1)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{\left(x - 2\right) \left(x + 1\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \left(- \frac{1}{3 \left(x + 1\right)} + \frac{1}{3 \left(x - 2\right)}\right)\, dx \]
    partial-fractionsRewrite the integrand as 1/3*(1/(x-2) - 1/(x+1))✓ Proved
  3. \[ = \frac{\ln{\left(x - 2 \right)}}{3} - \frac{\ln{\left(x + 1 \right)}}{3} \]
    antiderivativeIntegrate each term: ∫1/(x-2)dx = log(x-2), ∫1/(x+1)dx = log(x+1)✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.736

easy Video
Find \( \displaystyle \int \frac{1}{x \left(x + 3\right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \frac{1}{x \left(x + 3\right)}\, dx \]
    integralStart with the integral of the function.✓ Proved
  2. \[ = \int \left(- \frac{1}{3 \left(x + 3\right)} + \frac{1}{3 x}\right)\, dx \]
    partial-fractionsDecompose the integrand using partial fraction decomposition.✓ Proved
  3. \[ = \int \left(- \frac{1}{3 x + 9} + \frac{1}{3 x}\right)\, dx \]
    algebraDistribute the constant factor 1/3.✓ Proved
  4. \[ = \int \frac{1}{3 x}\, dx - \int \frac{1}{3 x + 9}\, dx \]
    linearitySplit the integral into two separate integrals.✓ Proved
  5. \[ = \frac{\int \frac{1}{x}\, dx}{3} - \frac{\int \frac{1}{x + 3}\, dx}{3} \]
    algebraFactor out the common constant 1/3.✓ Proved
  6. \[ = \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(x + 3 \right)}}{3} \]
    antiderivative algebraEvaluate the integrals using the natural logarithm. Factor out 1/3 to simplify the result.✓ Proved
Answer \( \frac{\ln{\left(x \right)} - \ln{\left(x + 3 \right)}}{3} + C \)
Top: the graph of F = log(x)/3 - log(x + 3)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/(x·(x + 3)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \frac{1}{x \left(x + 3\right)}\, dx \]
    integral✓ Proved
  2. \[ = \int \left(- \frac{1}{3 \left(x + 3\right)} + \frac{1}{3 x}\right)\, dx \]
    partial-fractions✓ Proved
  3. \[ = \frac{\int \frac{1}{x}\, dx}{3} - \frac{\int \frac{1}{x + 3}\, dx}{3} \]
    linearity✓ Proved
  4. \[ = \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(x + 3 \right)}}{3} \]
    antiderivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More partial fractions practice

Find \( \displaystyle \int \frac{1}{x^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.352easy Video✓ Every equation proved
Find \( \displaystyle \int \frac{x^{2} + 1}{x \left(x - 1\right)} \, dx \). (Omit the constant of integration.)
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Find \( \displaystyle \int \frac{1}{\left(x - 1\right) \left(x + 2\right)} \, dx \). (Omit the constant of integration.)
Problem 4.179medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{x \left(x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
Problem 4.185medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{x^{2} + 3 x} \, dx \). (Omit the constant of integration.)
Problem 4.191medium Video✓ Every equation proved
Find \( \displaystyle \int \frac{1}{9 x^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.231medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{x + 5}{x^{2} + x - 2} \, dx \). (Omit the constant of integration.)
Problem 4.353medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{3 x \left(3 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
Problem 4.368medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{\left(3 x - 2\right) \left(3 x + 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.372medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{9 x^{2} + 1}{3 x \left(3 x - 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.460medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{\left(3 x - 1\right) \left(3 x + 2\right)} \, dx \). (Omit the constant of integration.)
Problem 4.462medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{3 x}{\left(3 x + 1\right) \left(3 x + 3\right)} \, dx \). (Omit the constant of integration.)
Problem 4.551medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{\left(2 x - 1\right) \left(2 x + 2\right)} \, dx \). (Omit the constant of integration.)
Problem 4.553medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{3 x + 5}{9 x^{2} + 3 x - 2} \, dx \). (Omit the constant of integration.)
Problem 4.558medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{\left(x + 1\right) \left(x + 2\right)^{2}} \, dx \). (Omit the constant of integration.)
Problem 4.643medium Video✓ Every equation proved
Find \( \displaystyle \int \frac{1}{\left(2 x + 1\right) \left(2 x + 2\right)^{2}} \, dx \). (Omit the constant of integration.)
Problem 4.727medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{\left(3 x + 1\right) \left(3 x + 2\right)^{2}} \, dx \). (Omit the constant of integration.)
Problem 4.794medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{\left(2 x + 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.805medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{3 x + \left(x + 1\right)^{2} + 3} \, dx \). (Omit the constant of integration.)
Problem 4.928medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{6 x + \left(2 x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)
Problem 4.997medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{3 x + \left(x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)
Problem 4.1213medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{4 x^{2} + 6 x} \, dx \). (Omit the constant of integration.)
Problem 4.1216medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{1}{\left(x - 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.1247medium Video✓ Every equation proved
Find \( \displaystyle \int \frac{1}{\left(2 x - 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.1267medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{1}{\left(2 x - 2\right) \left(2 x + 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.1281medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{x + 4}{x + \left(x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)
Problem 4.1285medium Video✓ Every equation proved
Find \( \displaystyle \int \frac{1}{9 x + \left(3 x + 1\right)^{2} + 3} \, dx \). (Omit the constant of integration.)
Problem 4.1319medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{2 x}{\left(2 x + 1\right) \left(2 x + 3\right)} \, dx \). (Omit the constant of integration.)
Problem 4.1322medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{x - 1}{x \left(x + 2\right)} \, dx \). (Omit the constant of integration.)
Problem 4.1337medium Video✓ Every equation proved
Find \( \displaystyle \int \frac{1}{\left(3 x - 1\right)^{2} - 4} \, dx \). (Omit the constant of integration.)
Problem 4.1479medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{6 x - 1}{\left(3 x - 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
Problem 4.1481medium Video≈ Checked numerically
Find \( \displaystyle \int \frac{4 x + 1}{4 x^{2} - 1} \, dx \). (Omit the constant of integration.)
Problem 4.1491medium Video✓ Nihil obstat
Find \( \displaystyle \int \frac{\left(2 x - 1\right)^{2} + 1}{\left(2 x - 2\right) \left(2 x - 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.259hard Video≈ Checked numerically
Find \( \displaystyle \int \frac{6 x - 4}{- 2 x + \left(2 x - 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
Problem 4.471hard Video≈ Checked numerically
Find \( \displaystyle \int \frac{\left(3 x + 1\right)^{2} + 1}{3 x \left(3 x + 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.792hard Video✓ Nihil obstat
Find \( \displaystyle \int \frac{\left(2 x + 1\right)^{2} + 1}{2 x \left(2 x + 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.795hard Video≈ Checked numerically
Find \( \displaystyle \int \frac{\left(3 x - 1\right)^{2} + 1}{\left(3 x - 2\right) \left(3 x - 1\right)} \, dx \). (Omit the constant of integration.)
Problem 4.800hard Video✓ Nihil obstat
Find \( \displaystyle \int \frac{6 x + 2}{- 2 x + \left(2 x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
Problem 4.872hard Video≈ Checked numerically
Find \( \displaystyle \int \frac{9 x + 2}{- 3 x + \left(3 x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)
Problem 4.926hard Video✓ Nihil obstat
Find \( \displaystyle \int \frac{3 x + 4}{3 x + \left(3 x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)
Problem 4.1352hard Video≈ Checked numerically