## Math Equations

Calculus

$\int cf(x)dx=c\int f(x)dx$
$\int f(x)+g(x)dx=\int f(x)dx+\int g(x)dx$
$\int f(x)-g(x)dx=\int f(x)dx-\int g(x)dx$
$\int udv\, =uv-\int vdu$

$\int dx=x+C$
$\int adx=ax+C$
$\int x^ndx=\frac{1}{n+1}x^{n+1}+C$ if $n\neq -1$
$\int \frac{1}{x}dx=\ln |x|+C$
$\int \frac{1}{ax+b}dx=\frac{1}{a}\ln |ax+b|+C$ if $a\neq 0$

$\int \sin xdx=-\cos x+C$
$\int \cos xdx=\sin x+C$
$\int \tan xdx=\ln |\sec x|+C$
$\int \sin ^2xdx=\frac{1}{2}x-\frac{1}{4}\sin 2x+C$
$\int \cos ^2xdx=\frac{1}{2}x+\frac{1}{4}\sin 2x+C$
$\int \tan ^2xdx=\tan (x)-x+C$
$\int \sin ^nxdx=-\frac{\sin ^{n-1}x\cos x}{n}+\frac{n-1}{n}$
$\int \sin ^{n-2}xdx+C$ for $n>0$
$\int \cos ^nxdx=-\frac{\cos ^{n-1}x\sin x}{n}+\frac{n-1}{n}$
$\int \cos ^{n-2}xdx+C$ for $n>0$
$\int \tan ^nxdx=\frac{1}{(n-1)}\tan ^{n-1}x-\int \tan ^{n-2}xdx+C$ for $n\neq 1)$

$\int \sec xdx=\ln |\sec x+\tan x|+C=\ln \left|\tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right|+C$
$\int \csc xdx=-\ln |\csc x+\cot x|+C=\ln \left|\tan \left(\frac{1}{2}x\right)\right|+C$
$\int \cot xdx=\ln |\sin x|+C$
$\int \sec ^2kxdx=\frac{1}{k}\tan kx+C$
$\int \csc ^2kxdx=-\frac{1}{k}\cot kx+C$
$\int \cot ^2kxdx=-x-\frac{1}{k}\cot kx+C$
$\int \sec x\tan xdx=\sec x+C$
$\int \sec x\csc xdx=\ln |\tan x|+C$

$\int \sec ^nxdx=\frac{\sec ^{n-1}x\sin x}{n-1}+\frac{n-2}{n-1}$
$\int \sec ^{n-2}xdx+C$ for $n\neq 1$
$\int \csc ^nxdx=-\frac{\csc ^{n-1}x\cos x}{n-1}+\frac{n-2}{n-1}$
$\int \csc ^{n-2}xdx+C$ for $n\neq 1$
$\int \cot ^nxdx=-\frac{1}{n-1}\cot ^{n-1}x-\int \cot ^{n-2}xdx+C$ for $n\neq 1$
$\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin (x)+C$
$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin (x/a)+C$ if $a\neq 0$
$\int \frac{1}{1+x^2}dx=\arctan (x)+C$
$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan (x/a)+C$ if $a\neq 0$
$\int e^xdx=e^x+C$
$\int e^{ax}dx=\frac{1}{a}e^{ax}+C$ if $a\neq 0$
$\int a^xdx=\frac{1}{\ln a}a^x+C$ if $a>0,a\neq 1$
$\int \ln xdx=x\ln x-x+C$

$\int \arcsin (x)dx=x\arcsin (x)+\sqrt{1-x^2}+C$
$\int \arccos (x)dx=x\arccos (x)-\sqrt{1-x^2}+C$
$\int \arctan (x)dx=x\arctan (x)-\frac{1}{2}\ln (1+x^2)+C$

$\frac{d}{dx}(f+g)=\frac{df}{dx}+\frac{dg}{dx}$
$\frac{d}{dx}(cf)=c\frac{df}{dx}$
$\frac{d}{dx}(fg)=f\frac{dg}{dx}+g\frac{df}{dx}$
$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2}$

$\frac{d}{dx}(c)=0$
$\frac{d}{dx}x=1$
$\frac{d}{dx}x^n=nx^{n-1}$
$\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$
$\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$
$\frac{d}{dx}(c_nx^n+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_2x^2+c_1x+c_0)=nc_nx^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_2x+c_1$

$\frac{d}{dx}\sin (x)=\cos (x)$
$\frac{d}{dx}\cos (x)=-\sin (x)$
$\frac{d}{dx}\tan (x)=\sec ^2(x)$
$\frac{d}{dx}\cot (x)=-\csc ^2(x)$
$\frac{d}{dx}\sec (x)=\sec (x)\tan (x)\frac{d}{dx}\csc (x)=-\csc (x)\cot (x)$

$\frac{d}{dx}e^x=e^x\frac{d}{dx}a^x=a^x\ln (a)$ if $a>0$
$\frac{d}{dx}\ln (x)=\frac{1}{x}\frac{d}{dx}\log _a(x)=\frac{1}{x\ln (a)}$ if $a>0,a\neq 1$
${({f^g})^\prime } = {\left( {{e^{g\ln f}}} \right)^\prime } = {f^g}\left( {f'\frac{g}{f} + g'\ln f} \right),\qquad f > 0 \hfill$
${({c^f})^\prime } = {\left( {{e^{f\ln c}}} \right)^\prime } = f'{c^f}\ln c \hfill$

$\frac{d}{dx}\text{arcsinx}=\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\text{arccosx}=-\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\text{arctanx}=\frac{1}{1+x^2}$
$\frac{d}{dx}\text{arcsec}x=\frac{1}{|x|\sqrt{x^2-1}}$
$\frac{d}{dx}\text{arccot}x=\frac{-1}{1+x^2}$
$\frac{d}{dx}\text{arccsc}x=\frac{-1}{|x|\sqrt{x^2-1}}$

$\frac{d}{dx}\sinh x=\cosh x$
$\frac{d}{dx}\cosh x=\sinh x$
$\frac{d}{dx}\tanh x=\text{sech}^2x$
$\frac{d}{dx}\text{sech}x=-\tanh x\text{sech}x$
$\frac{d}{dx}\coth x=-\text{csch}^2x$
$\frac{d}{dx}\text{csch}x=-\coth x\text{csch}x$
$\frac{d}{dx}\sinh ^{-1}x=\frac{1}{\sqrt{x^2+1}}$
$\frac{d}{dx}\cosh ^{-1}x=\frac{-1}{\sqrt{x^2-1}}$
$\frac{d}{dx}\tanh ^{-1}x=\frac{1}{1-x^2}$
$\frac{d}{dx}\text{sech}^{-1}x=\frac{1}{x\sqrt{1-x^2}}$
$\frac{d}{dx}\coth ^{-1}x=\frac{-1}{1-x^2}$
$\frac{d}{dx}\text{csch}^{-1}x=\frac{-1}{|x|\sqrt{1+x^2}}$

$\sum _{i=N}^M f(i)=f(N)+f(N+1)+f(N+2)+\cdots +f(M)$

$\sum _{i=1}^n c=c+c+$$+c=nc,c\in \mathbb{R}$
$\sum _{i=1}^n i=1+2+3+$$+n=\frac{n(n+1)}{2}$
$\sum _{i=1}^n i^2=1^2+2^2+3^2+$$+n^2=\frac{n(n+1)(2n+1)}{6}$
$\sum _{i=1}^n i^3=1^3+2^3+3^3+$$+n^3=\frac{n^2(n+1)^2}{4}$

$\tan (x)=\frac{\sin x}{\cos x}$
$\sec (x)=\frac{1}{\cos x}$
$\cot (x)=\frac{\cos x}{\sin x}=\frac{1}{\tan x}$
$\csc (x)=\frac{1}{\sin x}$

$\sin ^2x+\cos ^2x=1\,$
$1 + {\tan ^2}(x) = {\sec ^2}x\; \hfill$
$1 + {\cot ^2}(x) = {\csc ^2}x\; \hfill$

$\sin (2x)=2\sin x\cos x\,$
$\cos (2x)=\cos ^2x-\sin ^2x\,$
$\tan (2x)=\frac{2\tan (x)}{1-\tan ^2(x)}$
$\cos ^2(x)=\frac{1+\cos (2x)}{2}$
$\sin ^2(x)=\frac{1-\cos (2x)}{2}$

$\sin (x+y)=\sin x\cos y+\cos x\sin y$
$\sin (x-y)=\sin x\cos y-\cos x\sin y$
$\cos (x+y)=\cos x\cos y-\sin x\sin y$
$\cos (x-y)=\cos x\cos y+\sin x\sin y$
$\sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$
$\sin x-\sin y=2\cos \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$
$\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$
$\cos x-\cos y=-2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$
$\tan x+\tan y=\frac{\sin (x+y)}{\cos x\cos y}$
$\tan x-\tan y=\frac{\sin (x-y)}{\cos x\cos y}$
$\cot x+\cot y=\frac{\sin (x+y)}{\sin x\sin y}$
$\cot x-\cot y=\frac{-\sin (x-y)}{\sin x\sin y}$

$\cos (x)\cos (y)=\frac{\cos (x+y)+\cos (x-y)}{2}$
$\sin (x)\sin (y)=\frac{\cos (x-y)-\cos (x+y)}{2}\,$
$\sin (x)\cos (y)=\frac{\sin (x+y)+\sin (x-y)}{2}\,$
$\cos (x)\sin (y)=\frac{\sin (x+y)-\sin (x-y)}{2}\,$