Undo the chain rule: spot the inside function and its derivative. 93 problems with worked solutions; in 87 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
Find \( \displaystyle \int 2 \left(2 x + 1\right) \sqrt{\left(2 x + 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
\[ \int \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(4 x + 2\right) \sqrt{4 x^{2} + 4 x + 2}\, dx \]
algebraExpand the expression inside the square root.✓ Proved
\[ = \int \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1}\, dx \]
rewritealgebraRewrite the radicand in terms of (2x+1)^2 + 1. Distribute the 2 into the linear term.✓ Proved
\[ = \frac{\left(\left(2 x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} \]
antiderivativeIntegrate using the power rule for the substitution u = (2x+1)^2 + 1.✓ Proved
Answer \( \frac{\left(\left(2 x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} + C \)
Top: the graph of F = ((2x + 1)² + 1)^(3/2)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (4x + 2)·sqrt((2x + 1)² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1}\, dx \]
integralrewriteRecognize the derivative of (2*x+1)**2+1✓ Proved
\[ = \frac{\left(\left(2 x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} \]
antiderivativesimplifyIntegrate using the substitution u=(2*x+1)**2+1 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
✓ 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
algebraRewrite the exponent using exponent laws.≈ Checked numerically
\[ = \int \frac{e^{2 x}}{e^{2 x} + e}\, dx \]
algebraalgebraSimplify the fraction by multiplying numerator and denominator by exp(1). Rewrite the integrand to reveal the derivative of the denominator.✓ Proved
\[ = \frac{\ln{\left(e^{2 x} + e \right)}}{2} \]
antiderivativeIntegrate using the substitution method or logarithmic rule.✓ Proved
\[ = \frac{\ln{\left(e e^{2 x - 1} + e \right)}}{2} \]
algebraRewrite the first term inside the log to match the original form.✓ Proved
algebraFactor out exp(1) from the expression inside the logarithm.✓ Proved
\[ = \frac{\ln{\left(e^{2 x - 1} + 1 \right)}}{2} \]
simplifyUse log properties to simplify the expression.≈ Checked numerically
Answer \( \frac{\ln{\left(e^{2 x - 1} + 1 \right)}}{2} + C \)
Top: the graph of F = log(e^(2x - 1) + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = e^(2x - 1)/(e^(2x - 1) + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{e^{2 x - 1}}{e^{2 x - 1} + 1}\, dx \]
integralrewriteFactor 1/2 out of the integrand.✓ Proved
\[ = \frac{\int \frac{2 e^{2 x - 1}}{e^{2 x - 1} + 1}\, dx}{2} \]
linearityrewritePull the constant 1/2 outside. Recognize that d/dx log(exp(2*x-1)+1) = 2*exp(2*x-1)/(exp(2*x-1)+1).✓ Proved
\[ = \frac{\ln{\left(e^{2 x - 1} + 1 \right)}}{2} \]
antiderivativeIntegrate the derivative to get the log.✓ Proved
✓ Nihil obstat Lines: 7 proved, 2 checked numerically. 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 \( \displaystyle \int e^{3 x - 1} \sin{\left(e^{3 x - 1} \right)} \, dx \). (Omit the constant of integration.)
\[ \int e^{3 x - 1} \sin{\left(e^{3 x - 1} \right)}\, dx \]
integralalgebraalgebraStart with the integral of the function. Multiply and divide by 3 to prepare for substitution. Distribute the constant.✓ Proved
\[ = \frac{\int 3 e^{3 x - 1} \sin{\left(e^{3 x - 1} \right)}\, dx}{3} \]
linearityrewritePull the constant out of the integral. Recognize the derivative of the inner function.✓ Proved
\[ = - \frac{\cos{\left(e^{3 x - 1} \right)}}{3} \]
antiderivativeIntegrate using the substitution rule.✓ Proved
Answer \( - \frac{\cos{\left(e^{3 x - 1} \right)}}{3} + C \)
Top: the graph of F = -cos(e^(3x - 1))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = e^(3x - 1)·sin(e^(3x - 1)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int e^{3 x - 1} \sin{\left(e^{3 x - 1} \right)}\, dx \]
integralalgebra✓ Proved
\[ = \frac{\int \sin{\left(e^{3 x - 1} \right)} \frac{d}{d x} e^{3 x - 1}\, dx}{3} \]
rewriteRewrite exp(3*x-1) as its derivative divided by 3.✓ Proved
\[ = - \frac{\cos{\left(e^{3 x - 1} \right)}}{3} \]
antiderivativesimplifyIntegral of sin(u) du is -cos(u).✓ 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 1/6 to match the derivative of the denominator.✓ Proved
\[ = \frac{\ln{\left(9 x^{2} + 1 \right)}}{6} \]
antiderivativeIntegral of 18*x/(9*x**2+1) is log(9*x**2+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
integralalgebraalgebrarewriteStart with the integral of the given function. Multiply and divide by 2 to prepare for substitution. Distribute the constant. Recognize the derivative of the inner function.✓ Proved
antiderivativeThe derivative of -cos(exp(2*x))/2 is exp(2*x)*sin(exp(2*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
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{2 x - 1}{x^{2} - x + 3}\, dx \]
algebrasimplifyrewritealgebraExpand the squared term. Combine like terms in the denominator. Recognize that the numerator is the derivative of the denominator. Compute the derivative of the denominator.✓ Proved
\[ = \ln{\left(x^{2} - x + 3 \right)} \]
antiderivativeIntegrate the expression using the logarithmic rule.✓ Proved
Answer \( \ln{\left(x^{2} - x + 3 \right)} + C \)
Top: the graph of F = log(x² - x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)/(x + (x - 1)² + 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
antiderivativeRecognize the numerator as the derivative of the denominator.✓ 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
rewritesubstitutionRewrite sin(x) as -diff(cos(x),x). Rearrange the integrand.✓ Proved
\[ = \frac{1}{\cos{\left(x \right)}} \]
antiderivativeIntegral of 1/cos(x)**2 * diff(cos(x),x) is 1/cos(x).✓ Proved
\[ = \sec{\left(x \right)} \]
simplify1/cos(x) is sec(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
✓ 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 \( \displaystyle \int \frac{3 x - 1}{\left(3 x - 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
\[ \int \frac{3 x - 1}{\left(3 x - 1\right)^{2} + 1}\, dx \]
integralrewriteStart with the integral of the given function. Rewrite the numerator to include the derivative of the denominator.✓ Proved
\[ = \frac{\int \frac{18 x - 6}{\left(3 x - 1\right)^{2} + 1}\, dx}{6} \]
linearityalgebraFactor out the constant 1/6. Rearrange the numerator to clearly show the derivative of the denominator.✓ Proved
\[ = \frac{\ln{\left(\left(3 x - 1\right)^{2} + 1 \right)}}{6} \]
antiderivativeApply the antiderivative rule for the form u'/u.✓ Proved
Answer \( \frac{\ln{\left(9 x^{2} - 6 x + 2 \right)}}{6} + C \)
Top: the graph of F = log(9x² - 6x + 2)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x - 1)/((3x - 1)² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{3 x - 1}{\left(3 x - 1\right)^{2} + 1}\, dx \]
integral✓ Proved
\[ = \frac{\int \frac{9 x - 3}{\left(3 x - 1\right)^{2} + 1}\, dx}{3} \]
algebraFactor out 1/3 to prepare for a logarithmic antiderivative.✓ Proved
\[ = \frac{\ln{\left(\left(3 x - 1\right)^{2} + 1 \right)}}{6} \]
antiderivativesimplifyIntegral of 3*(3*x-1)/((3*x-1)**2+1) is (1/2)*log((3*x-1)**2+1). Combine constants to give the final 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
antiderivativeIntegrate using the logarithmic rule for the derivative of the denominator.✓ Proved
Answer \( \frac{\ln{\left(4 x^{2} + 2 x + 3 \right)}}{2} + C \)
Top: the graph of F = log(4x² + 2x + 3)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (4x + 1)/(4x² + 2x + 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{4 x + 1}{4 x^{2} + 2 x + 3}\, dx \]
integral✓ Proved
\[ = \frac{\int \frac{8 x + 2}{4 x^{2} + 2 x + 3}\, dx}{2} \]
rewriteRewrite numerator as half of the derivative of the denominator.✓ Proved
antiderivativeIntegral of f'(x)/f(x) is log(f(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
linearityPull the constant 1/3 out of the integral.✓ Proved
\[ = \frac{e^{\left(x + 1\right)^{3}}}{3} \]
antiderivativeUse u = (x+1)^3, du = 3(x+1)^2 dx.✓ 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
integralalgebraalgebraStart with the integral of the given function. Multiply and divide by 2 to prepare for substitution. Rearrange the terms to highlight the derivative of the exponent.✓ Proved
\[ = \frac{e^{\left(x - 1\right)^{2}}}{2} \]
substitutionApply the substitution u = (x - 1)**2, where du = 2(x - 1)dx.✓ Proved
Answer \( \frac{e^{\left(x - 1\right)^{2}}}{2} + C \)
Top: the graph of F = e^((x - 1)²)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)·e^((x - 1)²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
linearitysubstitutionRecognize 2*(x-1) as the derivative of (x-1)**2✓ Proved
\[ = \frac{e^{\left(x - 1\right)^{2}}}{2} \]
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
Find \( \displaystyle \int 4 x^{2} e^{8 x^{3}} \, dx \). (Omit the constant of integration.)
\[ \int 4 x^{2} e^{8 x^{3}}\, dx \]
integralalgebraStart with the integral of the given function. Rewrite the coefficient to reveal the derivative of the exponent.✓ Proved
\[ = \frac{\int 24 x^{2} e^{8 x^{3}}\, dx}{6} \]
linearityPull the constant out of the integral.✓ Proved
\[ = \frac{e^{8 x^{3}}}{6} \]
antiderivativeThe integral of 24*x**2*exp(8*x**3) is exp(8*x**3).✓ Proved
Answer \( \frac{e^{8 x^{3}}}{6} + C \)
Top: the graph of F = e^(8x³)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 4x²·e^(8x³).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int 4 x^{2} e^{8 x^{3}}\, dx \]
integralrewriteRewrite 4 as (1/6)*24✓ Proved
\[ = \frac{\int 24 x^{2} e^{8 x^{3}}\, dx}{6} \]
linearityPull constant out of Integral✓ Proved
\[ = \frac{e^{8 x^{3}}}{6} \]
antiderivativeDerivative of exp(8*x**3) is 24*x**2*exp(8*x**3)✓ 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
antiderivativeIntegral of tan*u*sec*u is sec*u✓ 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 \( \displaystyle \int \frac{2 x + 1}{x^{2} + x + 3} \, dx \). (Omit the constant of integration.)
\[ \int \frac{2 x + 1}{x^{2} + x + 3}\, dx \]
integralrewriteStart with the given integral. Notice that the numerator is the derivative of the denominator.✓ Proved
\[ = \ln{\left(x^{2} + x + 3 \right)} \]
antiderivativeThe integral of u'/u is log(u).✓ Proved
Answer \( \ln{\left(x^{2} + x + 3 \right)} + C \)
Top: the graph of F = log(x² + x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)/(x² + x + 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \frac{2 x + 1}{x^{2} + x + 3}\, dx \]
integral✓ Proved
\[ = \ln{\left(x^{2} + x + 3 \right)} \]
antiderivativeNumerator is derivative of denominator.✓ 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
rewriteFactor out 1/3 to match the derivative of exp(3*x).✓ Proved
\[ = \frac{\ln{\left(e^{3 x} + 1 \right)}}{3} \]
antiderivativeIntegral of 1/u is log(u).✓ 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 \( \displaystyle \int \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2.0 x + 1.0\right) e^{\left(2 x + 1\right)^{2}}\, dx \]
algebraRewrite the integrand to expose the derivative of the exponent.✓ Proved
\[ = 0.25 e^{\left(2 x + 1\right)^{2}} \]
antiderivativeIntegrate the expression using the substitution rule.✓ Proved
Answer \( \frac{e^{\left(2 x + 1\right)^{2}}}{4} + C \)
Top: the graph of F = e^((2x + 1)²)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)·e^((2x + 1)²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}}\, dx \]
integral✓ Proved
\[ = \frac{\int \left(8 x + 4\right) e^{\left(2 x + 1\right)^{2}}\, dx}{4} \]
rewritesubstitutionFactor 4 to match the derivative of (2*x+1)**2 Recognize 4*(2*x+1) as the derivative of (2*x+1)**2✓ Proved
\[ = \frac{e^{\left(2 x + 1\right)^{2}}}{4} \]
antiderivativeIntegral of exp(u) du is exp(u)✓ 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 \( \displaystyle \int \left(3 x + 1\right) \cos{\left(\left(3 x + 1\right)^{2} \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(3 x + 1\right) \cos{\left(\left(3 x + 1\right)^{2} \right)}\, dx \]
integralalgebraStart with the integral of the given function. Rewrite the integrand to include the derivative of the inner function (3*x + 1)**2, which is 6*(3*x + 1) = 18*x + 6.✓ Proved
\[ = \frac{\sin{\left(\left(3 x + 1\right)^{2} \right)}}{6} \]
antiderivativeIntegrate the expression.✓ Proved
Answer \( \frac{\sin{\left(\left(3 x + 1\right)^{2} \right)}}{6} + C \)
Top: the graph of F = sin((3x + 1)²)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x + 1)·cos((3x + 1)²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(3 x + 1\right) \cos{\left(\left(3 x + 1\right)^{2} \right)}\, dx \]
integral✓ Proved
\[ = \frac{\int \left(18 x + 6\right) \cos{\left(\left(3 x + 1\right)^{2} \right)}\, dx}{6} \]
rewriteFactor out 1/6 to match the derivative of (3*x+1)**2✓ Proved
\[ = \frac{\sin{\left(\left(3 x + 1\right)^{2} \right)}}{6} \]
antiderivativeIntegral of f'(x)*cos(f(x)) is sin(f(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
Find \( \displaystyle \int 3 x \cos{\left(9 x^{2} \right)} \, dx \). (Omit the constant of integration.)
\[ \int 3 x \cos{\left(9 x^{2} \right)}\, dx \]
integralalgebraStart with the integral of the given function. Rewrite 3*x as (1/6)*18*x to prepare for the chain rule.✓ Proved
\[ = \frac{\sin{\left(9 x^{2} \right)}}{6} \]
antiderivativeThe antiderivative of 18*x*cos(9*x**2) is sin(9*x**2).✓ Proved
Answer \( \frac{\sin{\left(9 x^{2} \right)}}{6} + C \)
Top: the graph of F = sin(9x²)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 3x·cos(9x²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int 3 x \cos{\left(9 x^{2} \right)}\, dx \]
integral✓ Proved
\[ = \frac{\int 18 x \cos{\left(9 x^{2} \right)}\, dx}{6} \]
rewritesubstitutionrewriteFactor 3*x as (1/6)*18*x Recognize 18*x as derivative of 9*x**2 Use identity d/dx sin(u)=cos(u)*u'✓ Proved
\[ = \frac{\sin{\left(9 x^{2} \right)}}{6} \]
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