# 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. $\int {\frac {dx}{x^{2}+a^{2}}}$ ${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$ ${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$ ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises