# Calculus/Integration/Exercises

 ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises

## Integration of PolynomialsEdit

Evaluate the following:

1. $\int (x^2-2)^{2}\, dx$

$\frac{x^{5}}{5}-\frac{4x^{3}}{3}+4x+C$

2. $\int 8x^3\, dx$

$2x^{4}+C$

3. $\int (4x^2+11x^3)\, dx$

$\frac{4x^{3}}{3}+\frac{11x^{4}}{4}+C$

4. $\int (31x^{32}+4x^3-9x^4) \,dx$

$\frac{31x^{33}}{33}+x^{4}-\frac{9x^{5}}{5}+C$

5. $\int 5x^{-2}\, dx$

$-\frac{5}{x}+C$

Solutions

## Indefinite IntegrationEdit

Find the general antiderivative of the following:

6. $\int (\cos x+\sin x)\, dx$

$\sin x-\cos x+C$

7. $\int 3\sin x\, dx$

$-3\cos(x)+C$

8. $\int (1+\tan^2 x)\, dx$

$\tan x+C$

9. $\int (3x-\sec^2 x)\, dx$

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

10. $\int -e^x\, dx$

$-e^{x}+C$

11. $\int 8e^x\, dx$

$8e^{x}+C$

12. $\int \frac1{7x}\, dx$

$\frac{1}{7}\ln|x|+C$

13. $\int \frac1{x^2+a^2}\, dx$

$\frac{1}{a}\arctan\frac{x}{a}+C$

Solutions

## Integration by partsEdit

14. Consider the integral $\int \sin(x) \cos(x)\,dx$. Find the integral in two different ways. (a) Integrate by parts with $u=\sin(x)$ and $v' =\cos(x)$. (b) Integrate by parts with $u=\cos(x)$ and $v' =\sin(x)$. Compare your answers. Are they the same?

a. $\frac{\sin^{2}x}{2}$
b. $-\frac{\cos^{2}x}{2}$

Solutions

 ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises