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}\,

About russjohnson09

I am a student at GVSU.
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