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. $\int (\cos x+\sin x)\, dx$

7. $\int 3\sin x\, dx$

8. $\int (1+\tan^2 x)\, dx$

9. $\int (3x-\sec^2 x)\, dx$

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

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

12. $\int \frac1{7x}\, dx$

13. $\int \frac1{x^2+a^2}\, dx$

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?