1$\int{\frac{du}{u^2-a^2}}$
$= \frac{1}{2a}\ln\left|\frac{u - a}{u + a}\right|+C$
2$\int{\frac{du}{a^2-u^2}}$
$= \frac{1}{2a}\ln\left|\frac{u + a}{u - a}\right|+C$
3$\int\frac{du}{\sqrt{a^2-u^2}}$
$= \sin^{-1}\left(\frac{u}{a}\right)+C$
4$\int\frac{du}{u\sqrt{u^2 - a^2}}$
$= \frac{1}{a}\sec^{-1}\left(\frac{u}{a}\right)+C$
5$\int\frac{du}{a^2 + u^2}$
$= \frac{u}{a}\tan^{-1}\left(\frac{u}{a}\right)+C$
6$\sin(x + y)$
$= \sin(x)\cos(y)+\cos(x)sin(y)$
7$\sin(x - y)$
$= \sin(x)\cos(y)-\cos(x)\sin(y)$
8$\cos(x + y)$
$= \cos(x)\cos(y)-\sin(x)\sin(y)$
9$\cos(x - y)$
$= \cos(x)\cos(y)+\sin(x)\sin(y)$
10$\tan(x + y)$
$= \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)}$
11$\tan(x - y)$
$= \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)}$
12$\sin(2x)$
$= 2\sin(x)\cos(x)$
13$\cos(2x)$
$= \cos^2(x) - \sin^2(x)$
$= 2\cos^2(x) - 1$
$= 1 - 2sin^2(x)$
14$\tan(2x)$
$= \frac{2\tan(x)}{1 - \tan^2(x)}$
15$\sin^2(x)$
$= \frac{1 - \cos(2x)}{2}$
16$\cos^2(x)$
$= \frac{1 + \cos(2x)}{2}$
17$\int{b^udu}$
$\frac{b^u}{\ln{b}}+C$
18$\int\sin(u)du$
$= -\cos(u)+C$
19$\int\cos(u)du$
$= \sin(u)+C$
20$\int\sec^2(u)du$
$= \tan(u)+C$
21$\csc^2(u)du$
$= -\cot(u)+C$
22$\int\sec(u)\tan(u)du$
$= \sec(u)+C$
23$\int\csc(u)\cot(u)du$
$= -\csc(u)+C$
24$\int\tan(u)du$
$= \ln|\sec(u)|+C$
25$\int\cot(u)du$
$= \ln|\sin(u)|+C$
26$\int\sec(u)du$
$= \ln|\sec(u)+tan(u)|+C$
27$\int\csc(u)du$
$= \ln|\csc(u)-\cot(u)|+C$
28$\frac{d}{dx}(\arcsin(x))$
$= \frac{1}{\sqrt{1-x^2}}$
29$\frac{d}{dx}(\arccos(x))$
$= -\frac{1}{\sqrt{1-x^2}}$
30$\frac{d}{dx}(\arctan(x))$
$= \frac{1}{1+x^2}$
31$\frac{d}{dx}(\csc^{-1}(x))$
$= -\frac{1}{x\sqrt{x^2-1}}$
32$\frac{d}{dx}(\sec^{-1}(x))$
$= \frac{1}{x\sqrt{x^2-1}}$
33$\frac{d}{dx}(\cot^{-1}(x))$
$= -\frac{1}{1+x^2}$