Undo the product rule: ∫u dv = uv − ∫v du. 90 problems with worked solutions; in 82 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 \( \displaystyle \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x \cos{\left(4 x - 2 \right)} - \cos{\left(4 x - 2 \right)}\right)\, dx \]
linearityDistribute the cosine term.✓ Proved
\[ = 2 \int x \cos{\left(4 x - 2 \right)}\, dx - \int \cos{\left(4 x - 2 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} - 2 \int \frac{\sin{\left(4 x - 2 \right)}}{4}\, dx - \int \cos{\left(4 x - 2 \right)}\, dx \]
partsalgebrasimplifyApply integration by parts to the first integral, where u = x and dv = cos(4*x - 2)dx. Distribute the 2 into the parentheses. Simplify the coefficient of the first term.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} + \frac{\cos{\left(4 x - 2 \right)}}{8} - \int \cos{\left(4 x - 2 \right)}\, dx \]
antiderivativealgebraIntegrate the sine term. Simplify the sign and fraction.✓ Proved
\[ = \frac{x \sin{\left(4 x - 2 \right)}}{2} - \frac{\sin{\left(4 x - 2 \right)}}{4} + \frac{\cos{\left(4 x - 2 \right)}}{8} \]
antiderivativesimplifyIntegrate the final cosine term. Final simplification of the antiderivative.✓ Proved
Answer \( \frac{x \sin{\left(4 x - 2 \right)}}{2} - \frac{\sin{\left(4 x - 2 \right)}}{4} + \frac{\cos{\left(4 x - 2 \right)}}{8} + C \)
Top: the graph of F = x·sin(4x - 2)/2 - sin(4x - 2)/4 + cos(4x - 2)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x - 1)·cos(4x - 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x - 1\right) \cos{\left(4 x - 2 \right)}\, dx \]
integral✓ Proved
\[ = \frac{\int \left(4 x - 2\right) \cos{\left(4 x - 2 \right)}\, dx}{2} \]
algebraFactor 2*x-1 as (1/2)*(4*x-2)✓ Proved
\[ = \frac{\left(4 x - 2\right) \sin{\left(4 x - 2 \right)}}{8} - \frac{\int \sin{\left(4 x - 2 \right)}\, dx}{2} \]
partsIntegrate by parts with u=4*x-2, dv=cos(4*x-2)dx✓ Proved
\[ = \frac{\left(4 x - 2\right) \sin{\left(4 x - 2 \right)}}{8} + \frac{\cos{\left(4 x - 2 \right)}}{8} \]
antiderivativesimplifyIntegral of sin(4*x-2) is -cos(4*x-2)/4 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
Find \( \displaystyle \int \left(3 x + 1\right) \cos{\left(6 x + 2 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(3 x + 1\right) \cos{\left(6 x + 2 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(3 x \cos{\left(6 x + 2 \right)} + \cos{\left(6 x + 2 \right)}\right)\, dx \]
linearityDistribute the cosine term.✓ Proved
\[ = 3 \int x \cos{\left(6 x + 2 \right)}\, dx + \int \cos{\left(6 x + 2 \right)}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} - 3 \int \frac{\sin{\left(6 x + 2 \right)}}{6}\, dx + \int \cos{\left(6 x + 2 \right)}\, dx \]
partsApply integration by parts to the first integral with u=x and dv=cos(6x+2)dx.✓ Proved
\[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} - 3 \int \frac{\sin{\left(6 x + 2 \right)}}{6}\, dx \]
antiderivativeEvaluate the integral of cos(6x+2).✓ Proved
\[ = \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]
antiderivativesimplifysimplifyEvaluate the integral of sin(6x+2). Simplify the expression inside the parentheses. Distribute the 3 and simplify.✓ Proved
\[ = \left(\frac{x}{2} + \frac{1}{6}\right) \sin{\left(6 x + 2 \right)} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]
simplifyCombine the sine terms.✓ Proved
Answer \( \frac{x \sin{\left(6 x + 2 \right)}}{2} + \frac{\sin{\left(6 x + 2 \right)}}{6} + \frac{\cos{\left(6 x + 2 \right)}}{12} + C \)
Top: the graph of F = x·sin(6x + 2)/2 + sin(6x + 2)/6 + cos(6x + 2)/12. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x + 1)·cos(6x + 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(3 x + 1\right) \cos{\left(6 x + 2 \right)}\, dx \]
integral✓ Proved
\[ = \left(\frac{x}{2} + \frac{1}{6}\right) \sin{\left(6 x + 2 \right)} - \int \frac{\sin{\left(6 x + 2 \right)}}{2}\, dx \]
partsApply integration by parts with u = 3*x+1, dv = cos(6*x+2) dx✓ Proved
\[ = \left(\frac{x}{2} + \frac{1}{6}\right) \sin{\left(6 x + 2 \right)} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]
antiderivativeIntegrate sin(6*x+2) to get -(1/6)cos(6*x+2)✓ Proved
\[ = \frac{\left(6 x + 2\right) \sin{\left(6 x + 2 \right)}}{12} + \frac{\cos{\left(6 x + 2 \right)}}{12} \]
simplifyCombine terms over a common 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
antiderivativealgebraalgebrasimplifyEvaluate the integral of (x - 1)**(-2). Combine the terms by distributing the negative sign. Ensure the expression is simplified. Final simplified form of the antiderivative.✓ Proved
Top: the graph of F = (-log(x - 1) - 1)/(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = log(x - 1)/(x - 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
simplifyCombine terms over common 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
simplifyIntegrate the remaining exponential and combine 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
Find \( \displaystyle \int \left(3 x + 1\right)^{2} e^{3 x + 1} \, dx \). (Omit the constant of integration.)
\[ \int \left(3 x + 1\right)^{2} e^{3 x + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(9 x^{2} + 6 x + 1\right) e^{3 x + 1}\, dx \]
algebraExpand the squared binomial.✓ Proved
\[ = 6 \int x e^{3 x + 1}\, dx + 9 \int x^{2} e^{3 x + 1}\, dx + \int e^{3 x + 1}\, dx \]
linearityDistribute the exponential term and split the integral into three parts.✓ Proved
\[ = \frac{e^{3 x + 1}}{3} + 6 \int x e^{3 x + 1}\, dx + 9 \int x^{2} e^{3 x + 1}\, dx \]
antiderivativeEvaluate the integral of the constant term.✓ Proved
\[ = 3 x^{2} e^{3 x + 1} + \frac{e^{3 x + 1}}{3} - 9 \int \frac{2 x e^{3 x + 1}}{3}\, dx + 6 \int x e^{3 x + 1}\, dx \]
partsApply integration by parts to the first integral term.✓ Proved
\[ = 3 x^{2} e^{3 x + 1} + \frac{e^{3 x + 1}}{3} \]
simplifyalgebrasimplifySimplify the expression inside the parentheses. Distribute the 9 into the parentheses. Cancel the remaining integral terms.✓ Proved
Answer \( \frac{\left(9 x^{2} + 1\right) e^{3 x + 1}}{3} + C \)
Top: the graph of F = (9x² + 1)·e^(3x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x + 1)²·e^(3x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(3 x + 1\right)^{2} e^{3 x + 1}\, dx \]
integral✓ Proved
\[ = \frac{\left(3 x + 1\right)^{2} e^{3 x + 1}}{3} - \int \left(6 x + 2\right) e^{3 x + 1}\, dx \]
partsApply integration by parts once✓ Proved
\[ = \frac{\left(3 x + 1\right)^{2} e^{3 x + 1}}{3} - \frac{2 \left(3 x + 1\right) e^{3 x + 1}}{3} + 2 \int e^{3 x + 1}\, dx \]
partsApply integration by parts to the remaining integral✓ Proved
\[ = \frac{\left(3 x + 1\right)^{2} e^{3 x + 1}}{3} - \frac{\left(6 x + 2\right) e^{3 x + 1}}{3} + 2 \int e^{3 x + 1}\, dx \]
simplifyDistribute the factor 2 and simplify✓ Proved
\[ = \frac{\left(3 x + 1\right)^{2} e^{3 x + 1}}{3} - \frac{\left(6 x + 2\right) e^{3 x + 1}}{3} + \frac{2 e^{3 x + 1}}{3} \]
antiderivativeIntegrate exp(3*x + 1)✓ Proved
\[ = \frac{\left(- 6 x + \left(3 x + 1\right)^{2}\right) e^{3 x + 1}}{3} \]
simplifyFactor out exp(3*x + 1)/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
partsalgebraalgebraApply integration by parts to the second integral with u=log(x) and dv=x dx. Simplify the integrand inside the integral. Distribute the 2 into the parentheses.✓ 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 \( \displaystyle \int \left(2 x + 1\right) e^{- 2 x - 1} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x + 1\right) e^{- 2 x - 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x e^{- 2 x - 1} + e^{- 2 x - 1}\right)\, dx \]
linearityDistribute the exponential term.✓ Proved
\[ = 2 \int x e^{- 2 x - 1}\, dx + \int e^{- 2 x - 1}\, dx \]
linearitySplit into two integrals.✓ Proved
\[ = - x e^{- 2 x - 1} - 2 \int \left(- \frac{e^{- 2 x - 1}}{2}\right)\, dx + \int e^{- 2 x - 1}\, dx \]
partsApply integration by parts to the first integral with u=x and dv=exp(-2*x-1)dx.✓ Proved
\[ = - x e^{- 2 x - 1} + 2 \int e^{- 2 x - 1}\, dx \]
algebraalgebraalgebraSimplify the sign inside the parentheses. Distribute the constant 2. Combine the integral terms.✓ Proved
\[ = - x e^{- 2 x - 1} - e^{- 2 x - 1} \]
antiderivativealgebraEvaluate the remaining integral. Simplify the final expression.✓ Proved
Answer \( \left(- x - 1\right) e^{- 2 x - 1} + C \)
Top: the graph of F = (-x - 1)·e^(-2x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)·e^(-2x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x + 1\right) e^{- 2 x - 1}\, dx \]
integral✓ Proved
\[ = \frac{\left(- 2 x - 1\right) e^{- 2 x - 1}}{2} + \int e^{- 2 x - 1}\, dx \]
parts✓ Proved
\[ = \frac{\left(- 2 x - 1\right) e^{- 2 x - 1}}{2} - \frac{e^{- 2 x - 1}}{2} \]
antiderivative✓ Proved
\[ = \left(- x - 1\right) e^{- 2 x - 1} \]
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
Find \( \displaystyle \int 9 x^{2} \sin{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
\[ \int 9 x^{2} \sin{\left(3 x \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = 9 \int x^{2} \sin{\left(3 x \right)}\, dx \]
linearityPull out the constant factor 9.✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 9 \int \frac{2 x \cos{\left(3 x \right)}}{3}\, dx \]
partsApply integration by parts with u = x**2 and dv = sin(3*x)dx.✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 6 \int x \cos{\left(3 x \right)}\, dx \]
algebraSimplify the expression inside the integral.✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} - 6 \int \frac{\sin{\left(3 x \right)}}{3}\, dx \]
partsApply integration by parts again to the remaining integral.✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} - 2 \int \sin{\left(3 x \right)}\, dx \]
algebraDistribute the 2/3 factor.✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} + \frac{2 \cos{\left(3 x \right)}}{3} \]
antiderivativesimplifyEvaluate the integral of sin(3*x). Distribute the 9 and simplify the terms.✓ Proved
Answer \( - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} + \frac{2 \cos{\left(3 x \right)}}{3} + C \)
Top: the graph of F = -3x²·cos(3x) + 2x·sin(3x) + 2cos(3x)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 9x²·sin(3x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int 9 x^{2} \sin{\left(3 x \right)}\, dx \]
integral✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + \int 6 x \cos{\left(3 x \right)}\, dx \]
partsFirst integration by parts: u=x**2, dv=9*sin(3*x)dx✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} - \int 2 \sin{\left(3 x \right)}\, dx \]
partsSecond integration by parts: u=6*x, dv=cos(3*x)dx✓ Proved
\[ = - 3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} + \frac{2 \cos{\left(3 x \right)}}{3} \]
antiderivativeIntegrate the remaining sin 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
Answer \( x \sin{\left(x - 1 \right)} - \sin{\left(x - 1 \right)} + \cos{\left(x - 1 \right)} + C \)
Top: the graph of F = x·sin(x - 1) - sin(x - 1) + cos(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)·cos(x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
antiderivativeIntegrate sin(x-1) to get -cos(x-1) and 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
algebraFactor out sin(x + 1) to get the final form.✓ Proved
Answer \( x \sin{\left(x + 1 \right)} + \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} + C \)
Top: the graph of F = x·sin(x + 1) + sin(x + 1) + cos(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 1)·cos(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 \( \displaystyle \int \left(3 x - 1\right) e^{3 x - 1} \, dx \). (Omit the constant of integration.)
\[ \int \left(3 x - 1\right) e^{3 x - 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(3 x e^{3 x - 1} - e^{3 x - 1}\right)\, dx \]
linearityDistribute the exponential term.✓ Proved
\[ = 3 \int x e^{3 x - 1}\, dx - \int e^{3 x - 1}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = x e^{3 x - 1} - 3 \int \frac{e^{3 x - 1}}{3}\, dx - \int e^{3 x - 1}\, dx \]
partsApply integration by parts to the first integral term.✓ Proved
\[ = x e^{3 x - 1} - 2 \int e^{3 x - 1}\, dx \]
simplifyalgebraSimplify the expression. Combine the integral terms.✓ Proved
\[ = x e^{3 x - 1} - \frac{2 e^{3 x - 1}}{3} \]
antiderivativesimplifyEvaluate the remaining integral. Final simplification.✓ Proved
Answer \( \left(x - \frac{2}{3}\right) e^{3 x - 1} + C \)
Top: the graph of F = (x - 2/3)·e^(3x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x - 1)·e^(3x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(3 x - 1\right) e^{3 x - 1}\, dx \]
integral✓ Proved
\[ = \frac{\left(3 x - 1\right) e^{3 x - 1}}{3} - \int e^{3 x - 1}\, dx \]
partsApply integration by parts with u=3*x-1, dv=exp(3*x-1)dx✓ Proved
\[ = \frac{\left(3 x - 1\right) e^{3 x - 1}}{3} - \frac{e^{3 x - 1}}{3} \]
antiderivativeIntegrate exp(3*x-1) to get exp(3*x-1)/3✓ Proved
\[ = \frac{\left(3 x - 2\right) e^{3 x - 1}}{3} \]
simplifyCombine 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
✓ 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 \( \displaystyle \int 2 x \cos{\left(4 x \right)} \, dx \). (Omit the constant of integration.)
\[ \int 2 x \cos{\left(4 x \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = 2 \int x \cos{\left(4 x \right)}\, dx \]
linearityFactor out the constant 2.✓ Proved
\[ = \frac{x \sin{\left(4 x \right)}}{2} - 2 \int \frac{\sin{\left(4 x \right)}}{4}\, dx \]
partsalgebraApply integration by parts with u = x and dv = cos(4*x)dx. Distribute the x into the first term.✓ Proved
\[ = \frac{x \sin{\left(4 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} \]
antiderivativealgebrasimplifyEvaluate the integral of sin(4*x)/4. Simplify the signs and the denominator. Distribute the 2 and simplify the expression.✓ Proved
Answer \( \frac{x \sin{\left(4 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} + C \)
Top: the graph of F = x·sin(4x)/2 + cos(4x)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 2x·cos(4x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int 2 x \cos{\left(4 x \right)}\, dx \]
integral✓ Proved
\[ = \frac{x \sin{\left(4 x \right)}}{2} - \int \frac{\sin{\left(4 x \right)}}{2}\, dx \]
partsIntegrate by parts: u=2*x, dv=cos(4*x)dx✓ Proved
\[ = \frac{x \sin{\left(4 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} \]
✓ 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 = -x·cos(x + 1) + sin(x + 1) - cos(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 1)·sin(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
antiderivativeIntegral of cos(x+1) is sin(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
Find \( \displaystyle \int \left(2 x + 1\right) e^{2 x + 1} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x + 1\right) e^{2 x + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x e^{2 x + 1} + e^{2 x + 1}\right)\, dx \]
linearityDistribute the exponential term.✓ Proved
\[ = \int 2 x e^{2 x + 1}\, dx + \int e^{2 x + 1}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = \frac{e^{2 x + 1}}{2} + \int 2 x e^{2 x + 1}\, dx \]
antiderivativeIntegrate the second term.✓ Proved
\[ = x e^{2 x + 1} + \frac{e^{2 x + 1}}{2} - \int e^{2 x + 1}\, dx \]
partsApply integration by parts to the first term with u=x and dv=2*exp(2*x+1)dx.✓ Proved
\[ = x e^{2 x + 1} \]
antiderivativesimplifyIntegrate the remaining exponential term. Combine the terms.✓ Proved
Answer \( x e^{2 x + 1} + C \)
Top: the graph of F = x·e^(2x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 1)·e^(2x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x + 1\right) e^{2 x + 1}\, dx \]
integral✓ Proved
\[ = \left(x + \frac{1}{2}\right) e^{2 x + 1} - \int e^{2 x + 1}\, dx \]
parts✓ Proved
\[ = \frac{\left(2 x + 1\right) e^{2 x + 1}}{2} + \int \left(- e^{2 x + 1}\right)\, dx \]
rewriteRewrite the integral with a negative sign✓ Proved
\[ = \frac{\left(2 x + 1\right) e^{2 x + 1}}{2} - \frac{e^{2 x + 1}}{2} \]
antiderivative✓ Proved
\[ = x e^{2 x + 1} \]
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
integralsimplifypartsStart with the integral of the given function. The integrand is already in its simplest form. Use integration by parts with u = log(x - 1) and dv = (x - 1)**2 dx.✓ Proved
Top: the graph of F = -x³/9 + x²/3 - x/3 + (x³/3 - x² + x)·log(x - 1) - log(x - 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)²·log(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 \( \displaystyle \int \left(2 x + 1\right) \sin{\left(2 x + 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x + 1\right) \sin{\left(2 x + 1 \right)}\, dx \]
integralStart with the integral of the integrand.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} - \int \left(- \cos{\left(2 x + 1 \right)}\right)\, dx \]
partsalgebraApply integration by parts with u = 2*x + 1 and dv = sin(2*x + 1) dx. Simplify the integrand in the remaining integral.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} + \int \cos{\left(2 x + 1 \right)}\, dx \]
algebraDistribute the negative sign.✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} + \frac{\sin{\left(2 x + 1 \right)}}{2} \]
antiderivativeIntegrate the remaining cosine term.✓ Proved
\[ = \frac{\left(- 2 x - 1\right) \cos{\left(2 x + 1 \right)}}{2} + \frac{\sin{\left(2 x + 1 \right)}}{2} \]
algebraDistribute the fraction.✓ Proved
\[ = - x \cos{\left(2 x + 1 \right)} + \frac{\sin{\left(2 x + 1 \right)}}{2} - \frac{\cos{\left(2 x + 1 \right)}}{2} \]
algebraExpand the expression.✓ Proved
Answer \( - x \cos{\left(2 x + 1 \right)} + \frac{\sin{\left(2 x + 1 \right)}}{2} - \frac{\cos{\left(2 x + 1 \right)}}{2} + C \)
Top: the graph of F = -x·cos(2x + 1) + 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 = (2x + 1)·sin(2x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x + 1\right) \sin{\left(2 x + 1 \right)}\, dx \]
integral✓ Proved
\[ = - \frac{\left(2 x + 1\right) \cos{\left(2 x + 1 \right)}}{2} - \int \left(- \cos{\left(2 x + 1 \right)}\right)\, dx \]
partsApply integration by parts with u = 2*x+1, dv = sin(2*x+1)dx✓ Proved
\[ = \frac{\left(- 2 x - 1\right) \cos{\left(2 x + 1 \right)}}{2} + \int \cos{\left(2 x + 1 \right)}\, dx \]
simplifySimplify the product inside the integral✓ Proved
\[ = \frac{\left(- 2 x - 1\right) \cos{\left(2 x + 1 \right)}}{2} + \frac{\sin{\left(2 x + 1 \right)}}{2} \]
antiderivativeIntegrate cos(2*x+1) to get sin(2*x+1)/2✓ 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 = -x·cos(x - 1) + sin(x - 1) + cos(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x - 1)·sin(x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
antiderivativeIntegral of cos(x-1) is sin(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
Find \( \displaystyle \int \left(2 x - 1\right) \sin{\left(2 x - 1 \right)} \, dx \). (Omit the constant of integration.)
\[ \int \left(2 x - 1\right) \sin{\left(2 x - 1 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = - \frac{\left(2 x - 1\right) \cos{\left(2 x - 1 \right)}}{2} - \int \left(- \cos{\left(2 x - 1 \right)}\right)\, dx \]
partsApply integration by parts with u = 2*x - 1 and dv = sin(2*x - 1) dx.✓ Proved
\[ = - \frac{\left(2 x - 1\right) \cos{\left(2 x - 1 \right)}}{2} + \int \cos{\left(2 x - 1 \right)}\, dx \]
algebraSimplify the integrand in the remaining integral.✓ Proved
\[ = - \frac{\left(2 x - 1\right) \cos{\left(2 x - 1 \right)}}{2} + \frac{\sin{\left(2 x - 1 \right)}}{2} \]
antiderivativeEvaluate the integral of cos(2*x - 1).✓ Proved
\[ = - x \cos{\left(2 x - 1 \right)} + \frac{\sin{\left(2 x - 1 \right)}}{2} + \frac{\cos{\left(2 x - 1 \right)}}{2} \]
algebraDistribute the terms and simplify.✓ Proved
\[ = \left(\frac{1}{2} - x\right) \cos{\left(2 x - 1 \right)} + \frac{\sin{\left(2 x - 1 \right)}}{2} \]
simplifyCombine the cosine terms.✓ Proved
Answer \( - x \cos{\left(2 x - 1 \right)} + \frac{\sin{\left(2 x - 1 \right)}}{2} + \frac{\cos{\left(2 x - 1 \right)}}{2} + C \)
Top: the graph of F = -x·cos(2x - 1) + 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 = (2x - 1)·sin(2x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \int \left(2 x - 1\right) \sin{\left(2 x - 1 \right)}\, dx \]
integralStart with the given integral.✓ Proved
\[ = \frac{\left(1 - 2 x\right) \cos{\left(2 x - 1 \right)}}{2} + \int \cos{\left(2 x - 1 \right)}\, dx \]
partsIntegrate by parts: u=2*x-1, dv=sin(2*x-1)dx.✓ Proved
\[ = \frac{\left(1 - 2 x\right) \cos{\left(2 x - 1 \right)}}{2} + \frac{\sin{\left(2 x - 1 \right)}}{2} \]
\[ = - \frac{\left(2 x - 1\right) \cos{\left(2 x - 1 \right)}}{2} + \frac{\sin{\left(2 x - 1 \right)}}{2} \]
simplifyCombine terms over a common 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