Antiderivatives

Common Integrals

Simple Variable

Proper Integrals

Definite Integrals

See: Improper Integrals Integration Techniques

Exercises

1. Antiderivatives with Initial Conditions

Compute $F$, the antiderivative of the function $f(x)$ satisfying the following conditions:

1. $f(x) = \left(\frac{1}{x+1} - \frac{1}{x+2}\right)$, $F(0) = 0$
2. $f(x) = \frac{1}{x\ln x}$, $F(2) = 0$
3. $f(x) = \cos x(\sin x)^2$, $F(\pi) = 0$
4. $f(x) = 3e^{2x}$, $F(0) = 0$
5. $f(x) = \frac{\sin x \cos x}{(4 \sin^2 x - 1)}$, $F(\frac{\pi}{2})=0$
6. $f(x) = \frac{\sin x(\cos x)^2}{(\cos x)^3 + 1}$, $F(0) = 0$
7. $f(x) = \frac{e^x(3e^x-1)}{(e^x+1)(e^x-2)}$, $F(0) = 0$

2. Indefinite Integrals

Compute the following indefinite integrals:

1. $\int 2x^2e^{-x^3}dx$
2. $\int \frac{dt}{3+t^2}$
3. $\int \frac{x^2+x}{1+x^2}dx$
4. $\int \frac{\sin x}{1+\cos x}dx$
5. $\int \frac{e^x}{1+e^x}dx$
6. $\int x\sin 2x dx$
7. $\int x^3e^{-x}dx$
8. $\int (\ln x)^2dx$
9. $\int x^2\cos x dx$
10. $\int \frac{dx}{\sqrt{x}+x\sqrt{x}}$
11. $\int \frac{x+1}{x^2+4}dx$
12. $\int \frac{e^{3\sqrt{x}}}{\sqrt{x^2}}dx$
13. $\int x \sin x \cos x dx$
14. $\int x\arctan x dx$
15. $\int x^2\ln x dx$
16. $\int \frac{e^{5x}+e^{4x}-1}{e^{2x}}dx$
17. $\int \frac{x^2-1}{x^2+1}dx$
18. $\int \sin\sqrt{x}dx$
19. $\int e^x\sin x dx$
20. $\int \sqrt{\frac{1-x}{1+x}}dx$
21. $\int \frac{1}{4x^2+12x+9}dx$

3. True/False Questions

1. Consider the function $f(x) = \frac{x^2-3}{x-2}$: a) $F(x) = \frac{x^2}{2} + 2x + \ln(x-2)$ is an antiderivative of $f$ defined in $(2, +\infty)$

2. Consider the function $f(x) = \frac{\sqrt{x}-x^2}{x}$. Mark true or false: a) $F(x) = 2\sqrt{x} - \frac{1}{2}x^2$ is the antiderivative of $f$ such that $F(0) = 0$