Table of Integrals

Date: MATH 1100-2 - Spring 2002

In the following $a$, $b$ and $C$ are constants, and $u$, $v$ are functions.

1. $\int x^a dx = \frac{x^{a+1}}{a+1} + C$ for all $a\neq -1$.
2. $\int \frac{1}{x} dx = \ln(\vert x\vert) + C$.
3. $\int a^x dx = \frac{1}{\ln(a)} a^x + C$.
4. $\int e^x dx = e^x +C$.
5. $\int \frac{dx}{a^2-x^2} = \frac{1}{2a} \ln(\vert\frac{a+x}{a-x}\vert) + C$.
6. $\int \sqrt{a^2+x^2} dx = \frac{1}{2}(x\sqrt{a^2+x^2} + a^2 \ln(\vert x+\sqrt{a^2+x^2}\vert) + C$.
7. $\int \sqrt{x^2-a^2} dx = \frac{1}{2}(x\sqrt{x^2-a^2} - a^2 \ln(\vert x+\sqrt{a^2-x^2}\vert) + C$.
8. $\int \frac{dx}{\sqrt{a^2+x^2}} = \ln(\vert x+\sqrt{a^2+x^2}\vert) + C$.
9. $\int \frac{dx}{x\sqrt{a^2-x^2}} = -\frac{1}{a}\ln(\vert\frac{a+\sqrt{a^2-x^2}}{x}\vert) +C$.
10. $\int \frac{dx}{\sqrt{x^2-a^2}} = \ln(\vert x+\sqrt{x^2-a^2}\vert) + C$.
11. $\int \frac{dx}{x\sqrt{a^2+x^2}} = -\frac{1}{a}\ln(\vert\frac{a+\sqrt{a^2+x^2}}{x}\vert) +C$.
12. $\int \frac{xdx}{ax+b} = \frac{x}{a}-\frac{b}{a^2} \ln(\vert ax+b\vert) +C$.
13. $\int \frac{dx}{x(ax+b)} = \frac{1}{b}\ln(\vert\frac{x}{ax+b}\vert) +C$.
14. $\int \ln(x) dx = x (\ln(x) -1) +C$.
15. $\int\frac{xdx}{(ax+b)^2} = \frac{1}{a^2}(\ln(\vert ax+b\vert) + \frac{b}{ax+b})+C$.
16. $\int x\sqrt{ax+b} dx = \frac{2}{15a^2} (3ax-2b)(ax+b)^{\frac{3}{2}} + C$.
17. $\int u dv = uv - \int v du$.