# Calculus/Integration/Exercises

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

## Integration of Polynomials

Evaluate the following:

1. $\int (x^{2}-2)^{2}dx$
${\frac {x^{5}}{5}}-{\frac {4x^{3}}{3}}+4x+C$
${\frac {x^{5}}{5}}-{\frac {4x^{3}}{3}}+4x+C$
2. $\int 8x^{3}dx$
$2x^{4}+C$
$2x^{4}+C$
3. $\int (4x^{2}+11x^{3})dx$
${\frac {4x^{3}}{3}}+{\frac {11x^{4}}{4}}+C$
${\frac {4x^{3}}{3}}+{\frac {11x^{4}}{4}}+C$
4. $\int (31x^{32}+4x^{3}-9x^{4})dx$
${\frac {31x^{33}}{33}}+x^{4}-{\frac {9x^{5}}{5}}+C$
${\frac {31x^{33}}{33}}+x^{4}-{\frac {9x^{5}}{5}}+C$
5. $\int 5x^{-2}\,dx$
$-{\frac {5}{x}}+C$
$-{\frac {5}{x}}+C$

## Indefinite Integration

Find the general antiderivative of the following:

6. $\int {\bigl (}\cos(x)+\sin(x){\bigr )}dx$
$\sin(x)-\cos(x)+C$
$\sin(x)-\cos(x)+C$
7. $\int 3\sin(x)dx$
$-3\cos(x)+C$
$-3\cos(x)+C$
8. $\int {\bigl (}1+\tan ^{2}(x){\bigr )}dx$
$\tan(x)+C$
$\tan(x)+C$
9. $\int {\bigl (}3x-\sec ^{2}(x){\bigr )}dx$
${\frac {3x^{2}}{2}}-\tan(x)+C$
${\frac {3x^{2}}{2}}-\tan(x)+C$
10. $\int -e^{x}\,dx$
$-e^{x}+C$
$-e^{x}+C$
11. $\int 8e^{x}\,dx$
$8e^{x}+C$
$8e^{x}+C$
12. $\int {\frac {dx}{7x}}$
${\frac {\ln |x|}{7}}+C$
${\frac {\ln |x|}{7}}+C$
13. $\int {\frac {dx}{x^{2}+a^{2}}}$
${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$
${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$

## Integration by parts

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. $-{\frac {\cos ^{2}(x)}{2}}$
a. ${\frac {\sin ^{2}(x)}{2}}$
b. $-{\frac {\cos ^{2}(x)}{2}}$
 ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises