Calculus/Integration/Exercises

Evaluate the following:

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

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

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

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

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

Find the general antiderivative of the following:

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

7. ${\displaystyle \int 3\sin(x)dx}$

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

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

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

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

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

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

13. ${\displaystyle \int _{0}^{\pi /2}\sin(x)\cos(x)\,dx}$

14. ${\displaystyle \int _{0}^{\pi /4}\tan(x)\,dx}$.

15. ${\displaystyle \int _{1/2}^{1}{\frac {e^{\sqrt {2x-1}}}{\sqrt {2x-1}}}\,dx}$.

16. ${\displaystyle \int _{-3}^{-{\sqrt {6}}}{\frac {8x}{\sqrt {x^{2}-5}}}\,dx}$.

17. ${\displaystyle \int -{\frac {3}{2}}{\sqrt {\frac {2}{e^{3x-2}}}}\,dx}$.

18. ${\displaystyle \int {\frac {x\sec \left({\sqrt {x^{2}-5}}\right)\tan \left({\sqrt {x^{2}-5}}\right)}{50{\sqrt {x^{2}-5}}}}\,dx}$.

19. ${\displaystyle \int {\frac {2\sec ^{2}\left(\ln(x)\right)\tan \left(\ln(x)\right)}{x}}\,dx}$.

20. ${\displaystyle \int \left(e^{x}-1\right)x^{e^{x}-2}+x^{e^{x}-1}e^{x}\ln(x)\,dx}$.

21. ${\displaystyle \int 2\sec ^{2}\left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right){\frac {x^{3}-1}{x^{4}+2x}}\,dx}$.

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?

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