Reverse the derivative rules: powers, exponentials and trig functions. 82 problems with worked solutions; in 81 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. How we verify
antiderivativealgebraalgebraApply the power rule for integration to both terms. Simplify the fractions in the coefficients. Multiply the constants in each term.✓ Proved
Top: the graph of F = 2sqrt(3)·sqrt(x)·(9x² - 5)/15. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 3sqrt(3)·x^(3/2) - sqrt(3)/(3sqrt(x)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
antiderivativesimplifyIntegrate each power Combine constants✓ 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
antiderivativealgebrasimplifyalgebraApply the power rule for integration. Group the coefficients. Simplify 3**(1/2)/3 to 1/3**(1/2). Rewrite the inner term to match the original form.✓ Proved
\[ = \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} \]
algebraalgebrasimplifysimplifyDistribute the power to the numerator and denominator. Combine the denominator terms. Simplify 3**(1/2) * 3**(3/2) to 3**2. Final simplification of the coefficient.✓ Proved
Answer \( \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} + C \)
Top: the graph of F = 2(3x + 1)^(3/2)/9. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sqrt(3x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \sqrt{3 x + 1}\, dx \]
integral✓ Proved
\[ = \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} \]
antiderivativeApply the power rule for integrals.✓ 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
Find \( \displaystyle \int \frac{1}{\left(2 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
\[ \int \frac{1}{\left(2 x + 1\right)^{2}}\, dx \]
integralalgebraalgebraStart with the integral of the given function. Rewrite the negative exponent as a reciprocal. Factor out the coefficient of x from the base.✓ Proved
antiderivativeApply the power rule for integration.✓ Proved
\[ = - \frac{1}{4 x + 2} \]
algebraalgebraSimplify the expression. Multiply the denominator by 2 to clear the fraction inside.✓ Proved
Answer \( - \frac{1}{4 x + 2} + C \)
Top: the graph of F = -1/(4x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)^(-2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{1}{\left(2 x + 1\right)^{2}}\, dx \]
integral✓ Proved
\[ = \frac{\int \frac{2}{\left(2 x + 1\right)^{2}}\, dx}{2} \]
rewriterewrite✓ Proved
\[ = - \frac{1}{2 \left(2 x + 1\right)} \]
antiderivativesimplify✓ 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
antiderivativesimplifyIntegrate the expanded polynomial term. Combine all terms and simplify the expression.✓ Proved
Answer \( \frac{3 x \left(9 x^{3} - 12 x^{2} + 10 x - 4\right)}{4} + C \)
Top: the graph of F = 3x·(9x³ - 12x² + 10x - 4)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 6x + (3x - 1)³ - 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(6 x + \left(3 x - 1\right)^{3} - 2\right)\, dx \]
\[ = 3 x^{2} - 2 x + \int \left(3 x - 1\right)^{3}\, dx \]
antiderivative✓ Proved
\[ = 3 x^{2} - 2 x + \frac{\left(3 x - 1\right)^{4}}{12} \]
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
\[ = - 4 x^{2} + 5 x + 3 \int \left(2 x - 1\right)^{2}\, dx \]
antiderivativesimplifyIntegrate the linear term Simplify the quadratic term✓ Proved
\[ = - 4 x^{2} + 5 x + \frac{\left(2 x - 1\right)^{3}}{2} \]
antiderivativesimplifyIntegrate (2*x-1)**2 via substitution Simplify the result✓ 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
Find \( \displaystyle \int - 12 x + 3 \left(3 x + 1\right)^{2} - 3 \, dx \). (Omit the constant of integration.)
\[ \int \left(- 12 x + 3 \left(3 x + 1\right)^{2} - 3\right)\, dx \]
simplifyStart with the integral of the given expression.✓ Proved
\[ = \int \left(27 x^{2} + 6 x\right)\, dx \]
algebraalgebrasimplifyExpand the squared binomial. Distribute the 3 into the parentheses. Combine like terms.✓ Proved
\[ = \int 6 x\, dx + \int 27 x^{2}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = 9 x^{3} + 3 x^{2} \]
antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( x^{2} \left(9 x + 3\right) + C \)
Top: the graph of F = x²·(9x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = -12x + 3(3x + 1)² - 3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(- 12 x + 3 \left(3 x + 1\right)^{2} - 3\right)\, dx \]
integral✓ Proved
\[ = \int \left(27 x^{2} + 6 x\right)\, dx \]
simplifyExpand and combine like terms.✓ Proved
\[ = \int 6 x\, dx + \int 27 x^{2}\, dx \]
linearitySplit the integral.✓ Proved
\[ = 9 x^{3} + 3 x^{2} \]
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
✓ 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
algebraExpand (x-1)**2 and combine like terms.✓ Proved
\[ = x^{3} - 5 x^{2} + 8 x \]
antiderivativeIntegrate term by 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
linearitySplit the integral into three parts.✓ Proved
\[ = x^{3} - 2 x^{2} + x \]
antiderivativeIntegrate each term separately.✓ 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
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{2 x^{\frac{5}{2}}}{5} - 2 \sqrt{x} \]
antiderivativesimplifyIntegrate each power of x. Simplify the fractions.✓ 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
Top: the graph of F = 2sqrt(x - 1)·((x - 1)² - 5)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)^(3/2) - 1/sqrt(x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
Find \( \displaystyle \int \sqrt{3} \sqrt{x} \, dx \). (Omit the constant of integration.)
\[ \int \sqrt{3} \sqrt{x}\, dx \]
integralrewriteStart with the integral of the given function. Rewrite the square root as a fractional exponent.✓ Proved
\[ = \sqrt{3} \int \sqrt{x}\, dx \]
linearityPull the constant out of the integral.✓ Proved
\[ = \frac{2 \sqrt{3} x^{\frac{3}{2}}}{3} \]
antiderivativesimplifyApply the power rule for integration. Simplify the fraction by multiplying by the reciprocal.✓ Proved
Answer \( \frac{2 \sqrt{3} x^{\frac{3}{2}}}{3} + C \)
Top: the graph of F = 2sqrt(3)·x^(3/2)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sqrt(3)·sqrt(x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \sqrt{3} \sqrt{x}\, dx \]
integral✓ Proved
\[ = \sqrt{3} \int \sqrt{x}\, dx \]
rewriteRewrite sqrt(x) as x**(1/2) and factor out constants.✓ Proved
\[ = \frac{2 \sqrt{3} x^{\frac{3}{2}}}{3} \]
antiderivativesimplifyIntegrate x**(1/2) to get x**(3/2)/(3/2). Simplify the constant factor.✓ 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
algebraFactor out the constant 2 from the first term.✓ Proved
\[ = x^{2} + \frac{\ln{\left(x \right)}}{2} \]
antiderivativeEvaluate the integrals of x and 1/x.✓ Proved
Answer \( x^{2} + \frac{\ln{\left(x \right)}}{2} + C \)
Top: the graph of F = x² + log(x)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 2x + 1/(2x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x + \frac{1}{2 x}\right)\, dx \]
integral✓ Proved
\[ = \int \frac{1}{2 x}\, dx + \int 2 x\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = x^{2} + \int \frac{1}{2 x}\, dx \]
antiderivativeIntegrate 2*x to get x**2.✓ Proved
\[ = x^{2} + \frac{\int \frac{1}{x}\, dx}{2} \]
rewriteFactor 1/2 out of the second integral.✓ Proved
\[ = x^{2} + \frac{\ln{\left(x \right)}}{2} \]
antiderivativeIntegrate 1/x to get log(x).✓ 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
antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( \frac{x \left(x - 2\right) \left(x^{2} - 2 x + 6\right)}{4} + C \)
Top: the graph of F = x·(x - 2)·(x² - 2x + 6)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 2x + (x - 1)³ - 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
linearityPull the constant factor out of the integral.✓ Proved
\[ = \frac{3 x^{2}}{2} - x + \frac{\ln{\left(3 x - 1 \right)}}{3} \]
antiderivativeCompute the antiderivative of each term.✓ Proved
Answer \( \frac{3 x^{2}}{2} - x + \frac{\ln{\left(3 x - 1 \right)}}{3} + C \)
Top: the graph of F = 3x²/2 - x + log(3x - 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 3x - 1 + 1/(3x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(3 x - 1 + \frac{1}{3 x - 1}\right)\, dx \]
integral✓ Proved
\[ = \int \frac{1}{3 x - 1}\, dx + \int \left(3 x - 1\right)\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{3 x^{2}}{2} - x + \int \frac{1}{3 x - 1}\, dx \]
antiderivativeIntegrate the polynomial part.✓ Proved
\[ = \frac{3 x^{2}}{2} - x + \frac{\ln{\left(3 x - 1 \right)}}{3} \]
antiderivativeIntegrate 1/(3*x-1) using the logarithm.✓ 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
antiderivativeIntegrate each term using the power rule.✓ Proved
Answer \( \frac{x \left(x + 2\right) \left(x^{2} + 2 x + 6\right)}{4} + C \)
Top: the graph of F = x·(x + 2)·(x² + 2x + 6)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 2x + (x + 1)³ + 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
Find \( \displaystyle \int 8 x^{3} + 4 x \, dx \). (Omit the constant of integration.)
\[ \int \left(8 x^{3} + 4 x\right)\, dx \]
integralStart with the integral of the given polynomial.✓ Proved
\[ = 4 \int x\, dx + 8 \int x^{3}\, dx \]
linearitySplit the integral into two parts using linearity.✓ Proved
\[ = 2 x^{4} + 2 x^{2} \]
antiderivativesimplifyApply the power rule to each term. Simplify the coefficients.✓ Proved
Answer \( 2 x^{4} + 2 x^{2} + C \)
Top: the graph of F = 2x⁴ + 2x². A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 8x³ + 4x.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(8 x^{3} + 4 x\right)\, dx \]
integral✓ Proved
\[ = \int 4 x\, dx + \int 8 x^{3}\, dx \]
linearity✓ Proved
\[ = 2 x^{2} + \int 8 x^{3}\, dx \]
antiderivative✓ Proved
\[ = 2 x^{4} + 2 x^{2} \]
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
Find \( \displaystyle \int \sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(\sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)}\right)\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \sin{\left(3 x + 1 \right)}\, dx + \int \cos{\left(3 x + 1 \right)}\, dx \]
linearitySplit the integral into two parts using linearity.✓ Proved
\[ = - \frac{\cos{\left(3 x + 1 \right)}}{3} + \int \cos{\left(3 x + 1 \right)}\, dx \]
integralIntegrate the sine term.✓ Proved
\[ = \frac{\sin{\left(3 x + 1 \right)}}{3} - \frac{\cos{\left(3 x + 1 \right)}}{3} \]
integralIntegrate the cosine term.✓ Proved
Answer \( \frac{\sin{\left(3 x + 1 \right)} - \cos{\left(3 x + 1 \right)}}{3} + C \)
Top: the graph of F = sin(3x + 1)/3 - cos(3x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(3x + 1) + cos(3x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(\sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)}\right)\, dx \]
integral✓ Proved
\[ = - \frac{\cos{\left(3 x + 1 \right)}}{3} + \int \cos{\left(3 x + 1 \right)}\, dx \]
partsIntegrate sin(3*x+1) by substitution u=3*x+1.✓ Proved
\[ = \frac{\sin{\left(3 x + 1 \right)}}{3} - \frac{\cos{\left(3 x + 1 \right)}}{3} \]
✓ 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
Find \( \displaystyle \int \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(\sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right)\, dx \]
integralStart with the integral of the sum.✓ Proved
\[ = \int \sin{\left(2 x - 1 \right)}\, dx + \int \cos{\left(2 x - 1 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = - \frac{\cos{\left(2 x - 1 \right)}}{2} + \int \cos{\left(2 x - 1 \right)}\, dx \]
antiderivativeIntegrate the sine term.✓ Proved
\[ = \frac{\sin{\left(2 x - 1 \right)}}{2} - \frac{\cos{\left(2 x - 1 \right)}}{2} \]
antiderivativeIntegrate the cosine term.✓ Proved
Answer \( \frac{\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}}{2} + C \)
Top: the graph of F = sin(2x - 1)/2 - cos(2x - 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(2x - 1) + cos(2x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(\sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right)\, dx \]
integral✓ Proved
\[ = \int \sin{\left(2 x - 1 \right)}\, dx + \int \cos{\left(2 x - 1 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{\sin{\left(2 x - 1 \right)}}{2} - \frac{\cos{\left(2 x - 1 \right)}}{2} \]
antiderivativeIntegrate each part using ∫sin(ax+b)dx = -cos(ax+b)/a and ∫cos(ax+b)dx = sin(ax+b)/a.✓ 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
Top: the graph of F = 2sqrt(x + 1)·((x + 1)² - 5)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 1)^(3/2) - 1/sqrt(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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