Calculus/Integration/Exercises

Integration of Polynomials

Evaluate the following:

1. $\int (x^{2}-2)^{2}dx$

2. $\int 8x^{3}dx$

3. $\int (4x^{2}+11x^{3})dx$

4. $\int (31x^{32}+4x^{3}-9x^{4})dx$

5. $\int 5x^{-2}\,dx$

Indefinite Integration

Find the general antiderivative of the following:

6. $\int {\bigl (}\cos(x)+\sin(x){\bigr )}dx$

7. $\int 3\sin(x)dx$

8. $\int {\bigl (}1+\tan ^{2}(x){\bigr )}dx$

9. $\int {\bigl (}3x-\sec ^{2}(x){\bigr )}dx$

10. $\int -e^{x}\,dx$

11. $\int 8e^{x}\,dx$

12. $\int {\frac {dx}{7x}}$

13. $\int {\frac {dx}{x^{2}+a^{2}}}$

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?