# Calculus/Integration/Exercises

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

## Integration of Polynomials

Evaluate the following:

1. $\int (x^{2}-2)^{2}dx$ ${\frac {x^{5}}{5}}-{\frac {4x^{3}}{3}}+4x+C$ 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. $\int 5x^{-2}\,dx$ $-{\frac {5}{x}}+C$ ## Indefinite Integration

Find the general antiderivative of the following:

6. $\int {\bigl (}\cos(x)+\sin(x){\bigr )}dx$ $\sin(x)-\cos(x)+C$ 7. $\int 3\sin(x)dx$ $-3\cos(x)+C$ 8. $\int {\bigl (}1+\tan ^{2}(x){\bigr )}dx$ $\tan(x)+C$ 9. $\int {\bigl (}3x-\sec ^{2}(x){\bigr )}dx$ ${\frac {3x^{2}}{2}}-\tan(x)+C$ 10. $\int -e^{x}\,dx$ $-e^{x}+C$ 11. $\int 8e^{x}\,dx$ $8e^{x}+C$ 12. $\int {\frac {dx}{7x}}$ ${\frac {\ln |x|}{7}}+C$ 13. $\int {\frac {dx}{x^{2}+a^{2}}}$ ${\frac {\arctan {\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C$ ## 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. $-{\frac {\cos ^{2}(x)}{2}}$ ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises