1What is the integration by parts formula?
$\int u \, dv = uv - \int v \, du$
2What is the standard substitution used for $\int \sqrt{a^2 - x^2} \, dx$?
$x = a \sin A \quad \text{or} \quad x = a \cos A$
3What is the standard substitution used for $\int \sqrt{x^2 - a^2} \, dx$?
$x = a \sin A \quad \text{or} \quad x = a \cos A$
4What is the standard substitution used for $\int \sqrt{a^2 + x^2} \, dx$?
$x = a \tan A \quad \text{or} \quad x = a \cot A$
5What is the standard substitution used for $\int \sqrt{\frac{a \pm x}{a \mp x}} \, dx$?
$x = a \cos 2A$
6What is the standard substitution used for $\int {\frac{a \pm x}{a \mp x}} \, dx$?
$x = a \tan A$
7$\int (ax + b)^n \, dx$
$\frac{1}{a} \cdot \frac{1}{n+1} (ax + b)^{n+1}$ + C
8$\int a^x \, dx$
$\frac{a^x}{\ln a}$ + C
9$\int \alpha^{ax+b} \, dx$
$\frac{1}{a \ln \alpha} \alpha^{ax+b}$ + C
10$\int e^{ax+b} \, dx$
$\frac{1}{a} e^{ax+b}$ + C
11$\int \frac{1}{ax+b} \, dx$
$\frac{1}{a} \ln|ax+b|$ + C
12$\int \sin(ax+b) \, dx$
$-\frac{1}{a} \cos(ax+b)$ + C
13$\int \cos(ax+b) \, dx$
$\frac{1}{a} \sin(ax+b)$ + C
14$\int \sec^2(ax+b) \, dx$
$\frac{1}{a} \tan(ax+b)$ + C
15$\int \frac{1}{\sqrt{1 - (ax+b)^2}} \, dx$
$\frac{1}{a} \arcsin(ax+b)$ + C
16$\int \frac{-1}{\sqrt{1 - (ax+b)^2}} \, dx$
$\frac{1}{a} \arccos(ax+b)$ + C
17$\int \frac{1}{1 + (ax+b)^2} \, dx$
$\frac{1}{a} \arctan(ax+b)$ + C
18$\int \tan(ax+b) \, dx$
$-\frac{1}{a} \ln(\cos(ax+b))$ + C
19$\int \sinh(ax+b) \, dx$
$\frac{1}{a} \cosh(ax+b)$ + C
20$\int \cosh(ax+b) \, dx$
$\frac{1}{a} \sinh(ax+b)$ + C
21$\int \text{sech}^2(ax+b) \, dx$
$\frac{1}{a} \tanh(ax+b)$ + C
22$\int \frac{1}{\sqrt{1 + (ax+b)^2}} \, dx$
$\frac{1}{a} \sinh^{-1}(ax+b)$ + C
23$\int \frac{1}{\sqrt{(ax+b)^2 - 1}} \, dx$
$\frac{1}{a} \cosh^{-1}(ax+b)$ + C
24$\int \frac{1}{1 - (ax+b)^2} \, dx$
$\frac{1}{a} \tanh^{-1}(ax+b)$ + C