# 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. $\int {\frac {dx}{7x}}$
${\frac {\ln |x|}{7}}+C$
${\frac {\ln |x|}{7}}+C$

## Integration by Substitution

Find the anti-derivative or compute the integral depending on whether the integral is indefinite or definite.

13. $\int _{0}^{\pi /2}\sin(x)\cos(x)\,dx$
${\frac {1}{2}}$
${\frac {1}{2}}$
14. $\int _{0}^{\pi /4}\tan(x)\,dx$ .
${\frac {\ln(2)}{2}}$
${\frac {\ln(2)}{2}}$
15. $\int _{1/2}^{1}{\frac {e^{\sqrt {2x-1}}}{\sqrt {2x-1}}}\,dx$ .
$e-1$
$e-1$
16. $\int _{-3}^{-{\sqrt {6}}}{\frac {8x}{\sqrt {x^{2}-5}}}\,dx$ .
$-8$
$-8$
17. $\int -{\frac {3}{2}}{\sqrt {\frac {2}{e^{3x-2}}}}\,dx$ .
${\sqrt {\frac {2}{e^{3x-2}}}}+C$
${\sqrt {\frac {2}{e^{3x-2}}}}+C$
18. $\int {\frac {x\sec \left({\sqrt {x^{2}-5}}\right)\tan \left({\sqrt {x^{2}-5}}\right)}{50{\sqrt {x^{2}-5}}}}\,dx$ .
${\frac {\sec \left({\sqrt {x^{2}-5}}\right)}{50}}+C$
${\frac {\sec \left({\sqrt {x^{2}-5}}\right)}{50}}+C$
19. $\int {\frac {2\sec ^{2}\left(\ln(x)\right)\tan \left(\ln(x)\right)}{x}}\,dx$ .
$\tan ^{2}\left(\ln(x)\right)+C$
$\tan ^{2}\left(\ln(x)\right)+C$
20. $\int \left(e^{x}-1\right)x^{e^{x}-2}+x^{e^{x}-1}e^{x}\ln(x)\,dx$ .
$x^{e^{x}-1}+C$
$x^{e^{x}-1}+C$
21. $\int 2\sec ^{2}\left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right){\frac {x^{3}-1}{x^{4}+2x}}\,dx$ .
$\tan \left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right)+C$
$\tan \left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right)+C$

## Integration by parts

30. 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 by Trigonometric Substitution

40. $\int {\frac {dx}{x^{2}+a^{2}}}$
${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$
${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$
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