u-Substitution — Practice Problems and Worked Solutions

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18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the integral using substitution.

   $$\int 2x{\left(x^2+1\right)}^{3}dx$$

   Answer: $\frac{x^8}{4}+x^6+\frac{3x^4}{2}+x^2+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int 2x{\left(x^2+1\right)}^{3}dx=\frac{x^8}{4}+x^6+\frac{3x^4}{2}+x^2+C$$
2. 2.

   Evaluate the integral using substitution.

   $$\int 3x^2{e}^{x^3}dx$$

   Answer: $e^{x^3}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int 3x^2{e}^{x^3}dx=e^{x^3}+C$$
3. 3.

   Evaluate the integral using substitution.

   $$\int \frac{2x}{x^2+4}dx$$

   Answer: $\log \left(x^2+4\right)+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int \frac{2x}{x^2+4}dx=\log \left(x^2+4\right)+C$$
4. 4.

   Evaluate the integral using substitution.

   $$\int e^{\sin \left(x\right)}\cos \left(x\right)dx$$

   Answer: $e^{\sin \left(x\right)}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int e^{\sin \left(x\right)}\cos \left(x\right)dx=e^{\sin \left(x\right)}+C$$
5. 5.

   Evaluate the integral using substitution.

   $$\int 2x\cos \left(x^2\right)dx$$

   Answer: $\sin \left(x^2\right)+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int 2x\cos \left(x^2\right)dx=\sin \left(x^2\right)+C$$
6. 6.

   Evaluate the integral using substitution.

   $$\int {\left(2x+1\right)}^{5}dx$$

   Answer: $\frac{x\left(16x^5+48x^4+60x^3+40x^2+15x+3\right)}{3}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int {\left(2x+1\right)}^{5}dx=\frac{16x^6}{3}+16x^5+20x^4+\frac{40x^3}{3}+5x^2+x+C$$
7. 7.

   Evaluate the integral using substitution.

   $$\int \frac{x}{\sqrt{x^2+1}}dx$$

   Answer: $\sqrt{x^2+1}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int \frac{x}{\sqrt{x^2+1}}dx=\sqrt{x^2+1}+C$$
8. 8.

   Evaluate the integral using substitution.

   $$\int \sin \left(x\right){\cos }^3\left(x\right)dx$$

   Answer: $-\frac{{\cos }^4\left(x\right)}{4}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int \sin \left(x\right){\cos }^3\left(x\right)dx=-\frac{{\cos }^4\left(x\right)}{4}+C$$
9. 9.

   Evaluate the integral using substitution.

   $$\int 2x{\left(x^2+3\right)}^{4}dx$$

   Answer: $\frac{x^2\left(x^8+15x^6+90x^4+270x^2+405\right)}{5}+C$

   Step-by-step solution

   1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
   2. $$\int 2x{\left(x^2+3\right)}^{4}dx=\frac{x^{10}}{5}+3x^8+18x^6+54x^4+81x^2+C$$
10. 10.

    Evaluate the integral using substitution.

    $$\int 3x^2{\left(x^3+1\right)}^{2}dx$$

    Answer: $\frac{x^9}{3}+x^6+x^3+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int 3x^2{\left(x^3+1\right)}^{2}dx=\frac{x^9}{3}+x^6+x^3+C$$
11. 11.

    Evaluate the integral using substitution.

    $$\int \frac{2x}{x^2+9}dx$$

    Answer: $\log \left(x^2+9\right)+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int \frac{2x}{x^2+9}dx=\log \left(x^2+9\right)+C$$
12. 12.

    Evaluate the integral using substitution.

    $$\int e^{\cos \left(x\right)}\sin \left(x\right)dx$$

    Answer: $-e^{\cos \left(x\right)}+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int e^{\cos \left(x\right)}\sin \left(x\right)dx=-e^{\cos \left(x\right)}+C$$
13. 13.

    Evaluate the integral using substitution.

    $$\int 2x\sin \left(x^2\right)dx$$

    Answer: $-\cos \left(x^2\right)+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int 2x\sin \left(x^2\right)dx=-\cos \left(x^2\right)+C$$
14. 14.

    Evaluate the integral using substitution.

    $$\int {\left(3x+2\right)}^{4}dx$$

    Answer: $\frac{x\left(81x^4+270x^3+360x^2+240x+80\right)}{5}+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int {\left(3x+2\right)}^{4}dx=\frac{81x^5}{5}+54x^4+72x^3+48x^2+16x+C$$
15. 15.

    Evaluate the integral using substitution.

    $$\int x{e}^{x^2}dx$$

    Answer: $\frac{e^{x^2}}{2}+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int x{e}^{x^2}dx=\frac{e^{x^2}}{2}+C$$
16. 16.

    Evaluate the integral using substitution.

    $$\int {\sin }^2\left(x\right)\cos \left(x\right)dx$$

    Answer: $\frac{{\sin }^3\left(x\right)}{3}+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int {\sin }^2\left(x\right)\cos \left(x\right)dx=\frac{{\sin }^3\left(x\right)}{3}+C$$
17. 17.

    Evaluate the integral using substitution.

    $$\int {\left(4x-1\right)}^{3}dx$$

    Answer: $x\left(16x^3-16x^2+6x-1\right)+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int {\left(4x-1\right)}^{3}dx=16x^4-16x^3+6x^2-x+C$$
18. 18.

    Evaluate the integral using substitution.

    $$\int \frac{x}{{\left(x^2+1\right)}^2}dx$$

    Answer: $-\frac{1}{2x^2+2}+C$

    Step-by-step solution

    1. $$\text{Choose the inner expression as }u\text{ and match its differential.}$$
    2. $$\int \frac{x}{{\left(x^2+1\right)}^2}dx=-\frac{1}{2x^2+2}+C$$

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