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Calculus 2 : Solving Integrals by Substitution

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Example Questions

Example Question #981 : Integrals

Solve: $\int10e^{10x}$ .

Possible Answers:

10e^x + C

$e^{10x}+C$

$\frac{e^x}{10} + C$

e^x + C

Correct answer:

$e^{10x}+C$

Explanation:

Substitute u=10x, du = 10 dx :

$\int e^{u}du$ ,

which is equal to

$e^{u}+ C$ .

Replace u with 10x:

$e^{10x}+C$ .

Report an Error

Example Question #982 : Integrals

$\int cos^3xsinxdx$

Possible Answers:

$\frac{cos^2x}{2} +C$

$\frac{sin^4x}{4}$

$\frac{-cos^4x}{4} + C$

cos^4x+C

$\frac{cos^4x}{4} + C$

Correct answer:

$\frac{-cos^4x}{4} + C$

Explanation:

Substitute u=cosx, du=-sinxdx :

$\int- u^3du=-\frac{u^4}{4} + C$ .

Replace u=cos x :

$\frac{-cos^4x}{4} + C$ .

Report an Error

Example Question #983 : Integrals

Solve $\int sin^3x cos x dx$ .

Possible Answers:

$\frac{sin^2x}{2} +C$

3sin^2x +C

sin^4x +C

cos^4x+C

$\frac{sin^4x}{4} +C$

Correct answer:

$\frac{sin^4x}{4} +C$

Explanation:

Substitute u=sin x, du=cosx :

$\int u^3 du=\frac{u^4}{4} +C$ .

Replace u=sin x :

$\frac{sin^4x}{4} +C$ .

Report an Error

Example Question #984 : Integrals

Solve $\int 5e^{5x}dx$ .

Possible Answers:

$e^{4x}+C$

$\frac{e^{6x}}{6} + C$

$e^{5x} + C$

$5e^{x} + C$

$5e^{4x} +C$

Correct answer:

$e^{5x} + C$

Explanation:

Substitue u=5x, du=5 .

$\int e^udu=e^u+C$ .

Replace u=5x :

$e^{5x} + C$ .

Report an Error

Example Question #985 : Integrals

Find $\int \frac{dx}{(x+2)^4}$ .

Possible Answers:

$\frac{-(x+2)^{-5}}{5}+C$

ln((x+2)^4)+C

$\frac{-(x+2)^{-3}}{3}+C$

${(x+2)^{-4}}+C$

4ln(x+2)+C

Correct answer:

$\frac{-(x+2)^{-3}}{3}+C$

Explanation:

Substitute u=x+2, du=dx :

$\int \frac{1}{u^4}du=-\frac{u^{-3}}{3}$ .

Replace u=x+2 :

$\frac{-(x+2)^{-3}}{3}+C$

Report an Error

Example Question #986 : Integrals

Solve $\int \frac{x^2}{x^3+1}$ .

Possible Answers:

$ln\left | x+1\right | + C$

$\frac{1}{3}ln\left | x^3+1\right | + C$

$ln\left | x^3+1\right | + C$

$\frac{x^4}{4}+\frac{x^3}{3}+C$

Correct answer:

$\frac{1}{3}ln\left | x^3+1\right | + C$

Explanation:

Substitute u=x^3 + 1, du=3x^2 dx .

$\int \frac{1}{3}u^{-1}du=\frac{1}{3}ln\left | u \right |+C$ .

Replace u=x^3+1 :

$\frac{1}{3}ln\left | x^3+1\right | + C$ .

Report an Error

Example Question #51 : Solving Integrals By Substitution

$\int xe^{x^2}$ .

Possible Answers:

$\frac{1}{2}e^{x^2}$

$e^{2x}$

$\frac{1}{2}e^{x}$

$e^{x^2}$

$\frac{1}{2}e^{2x}$

Correct answer:

$\frac{1}{2}e^{x^2}$

Explanation:

Substitute u=x^2, du=2x :

$\frac{1}{2} \int e^udu=\frac{1}{2}e^u$ .

Replace u=x^2 :

$\frac{1}{2}e^{x^2}$ .

Report an Error

Example Question #52 : Solving Integrals By Substitution

Solve $\int (x^2+x)^9(2x+1)dx$ .

Possible Answers:

2x^2+x+C

$\frac{(x^2+x)^{10}}{10}+C$

$10{(x^2+x)^{9}}+C$

x^2+x

${(x^2+x)^{10}}+C$

Correct answer:

$\frac{(x^2+x)^{10}}{10}+C$

Explanation:

Substitute u=x^2+x, du=2x+1 .

$\int u^9du=\frac{u^{10}}{10}$ .

Replace u=x^2+x :

$\frac{(x^2+x)^{10}}{10}+C$ .

Report an Error

Example Question #993 : Integrals

Solve $\int x^3(x^4+16)^6dx$ .

Possible Answers:

$\frac{(x^4+16)^4}{4}$

$\frac{(x^4+16)^7}{28}$

(x^4+16)^7

$\frac{(x^4+16)^7}{7}$

$\frac{(x^4+16)^2}{8}$

Correct answer:

$\frac{(x^4+16)^7}{28}$

Explanation:

Let u=x^4+16, du=4x^3 :

$\int \frac{1}{4}u^6du=\frac{1}{4}*\frac{u^7}{7}=\frac{u^7}{28}$ .

Replace u=x^4+16 :

$\frac{(x^4+16)^7}{28}$ .

Report an Error

Example Question #994 : Integrals

Solve $\int sin^3xcosxdx$ .

Possible Answers:

${sin^4x}$

$\frac{sinx}{4}$

3sinxcosx

$\frac{cos^4x}{4}$

$\frac{sin^4x}{4}$

Correct answer:

$\frac{sin^4x}{4}$

Explanation:

Substitute u=sin x, du=cos x :

$\int u^3 du=\frac{u^4}{4}$ .

Replace u=sin x :

$\frac{sin^4x}{4}$ .

Report an Error

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Calculus 2 : Solving Integrals by Substitution

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #981 : Integrals

Example Question #982 : Integrals

Example Question #983 : Integrals

Example Question #984 : Integrals

Example Question #985 : Integrals

Example Question #986 : Integrals

Example Question #51 : Solving Integrals By Substitution

Example Question #52 : Solving Integrals By Substitution

Example Question #993 : Integrals

Example Question #994 : Integrals

All Calculus 2 Resources

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