# Calculus/Integration/Exercises

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

## Integration of Polynomials

Evaluate the following:

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

## Indefinite Integration

Find the general antiderivative of the following:

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

## Integration by Substitution

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

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

## Integration by parts

30. Consider the integral ${\displaystyle \int \sin(x)\cos(x)dx}$  . Find the integral in two different ways. (a) Integrate by parts with ${\displaystyle u=\sin(x)}$  and ${\displaystyle v'=\cos(x)}$  . (b) Integrate by parts with ${\displaystyle u=\cos(x)}$  and ${\displaystyle v'=\sin(x)}$  . Compare your answers. Are they the same?
a. ${\displaystyle {\frac {\sin ^{2}(x)}{2}}}$
b. ${\displaystyle -{\frac {\cos ^{2}(x)}{2}}}$
a. ${\displaystyle {\frac {\sin ^{2}(x)}{2}}}$
b. ${\displaystyle -{\frac {\cos ^{2}(x)}{2}}}$

## Integration by Trigonometric Substitution

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