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Table of Standard Forms

$\dfrac{dy}{dx}$	$y$	$\int y\, dx$
Algebraic.
$1$	$x$	$\frac{1}{2} x^2 + C$
$0$	$a$	$ax + C$
$1$	$x ± a$	$\frac{1}{2} x^2 ± ax + C$
$a$	$ax$	$\frac{1}{2} ax^2 + C$
$2x$	$x^2$	$\frac{1}{3} x^3 + C$
$nx^{n-1}$	$x^n$	$\dfrac{1}{n+1} x^{n+1} + C$
$-x^{-2}$	$x^{-1}$	$\log_\epsilon x + C$
$\dfrac{du}{dx} ± \dfrac{dv}{dx} ± \dfrac{dw}{dx}$	$u ± v ± w$	$\int u\, dx ± \int v\, dx ± \int w\, dx$
$u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}$	$uv$	No general form known
$\dfrac{v\, \dfrac{du}{dx} - u\, \dfrac{dv}{dx}}{v^2}$	$\dfrac{u}{v}$	No general form known
$\dfrac{du}{dx}$	$u$	$ux - \int x\, du + C$
Exponential and Logarithmic.
$\epsilon^x$	$\epsilon^x$	$\epsilon^x + C$
$x^{-1}$	$\log_\epsilon x$	$x(\log_\epsilon x - 1) + C$
$0.4343 × x^{-1}$	$\log_{10} x$	$0.4343x (\log_\epsilon x - 1) + C$
$a^x \log_\epsilon a$	$a^x$	$\dfrac{a^x}{\log_\epsilon a} + C$
Trigonometrical.
$\cos x$	$\sin x$	$-\cos x + C$
$-\sin x$	$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x$	$-\log_\epsilon \cos x + C$
Circular (Inverse).
$\dfrac{1}{\sqrt{(1-x^2)}}$	$\arcsin x$	$x · \arcsin x + \sqrt{1 - x^2} + C$
$-\dfrac{1}{\sqrt{(1-x^2)}}$	$\arccos x$	$x · \arccos x - \sqrt{1 - x^2} + C$
$\dfrac{1}{1+x^2}$	$\arctan x$	$x · \arctan x - \frac{1}{2} \log_\epsilon (1 + x^2) + C$
Hyperbolic.
$\cosh x$	$\sinh x$	$\cosh x + C$
$\sinh x$	$\cosh x$	$\sinh x + C$
$\text{sech}^2 x$	$\tanh x$	$\log_\epsilon \cosh x + C$
Miscellaneous.
$-\dfrac{1}{(x + a)^2}$	$\dfrac{1}{x + a}$	$\log_\epsilon (x+a) + C$
$-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}$	$\dfrac{1}{\sqrt{a^2 + x^2}}$	$\log_\epsilon (x + \sqrt{a^2 + x^2}) + C$
$\mp \dfrac{b}{(a ± bx)^2}$	$\dfrac{1}{a ± bx}$	$± \dfrac{1}{b} \log_\epsilon (a ± bx) + C$
$-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}$	$\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}$	$\dfrac{x}{\sqrt{a^2 + x^2}} + C$
$a · \cos ax$	$\sin ax$	$-\dfrac{1}{a} \cos ax + C$
$-a · \sin ax$	$\cos ax$	$\dfrac{1}{a} \sin ax + C$
$a · \sec^2ax$	$\tan ax$	$-\dfrac{1}{a} \log_\epsilon \cos ax + C$
$\sin 2x$	$\sin^2 x$	$\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$
$-\sin 2x$	$\cos^2 x$	$\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$
$n · \sin^{n-1} x · \cos x$	$\sin^n x$	$-\frac{\cos x}{n} \sin^{n-1} x + \frac{n-1}{n} \int \sin^{n-2} x\, dx + C$
$-\dfrac{\cos x}{\sin^2 x}$	$\dfrac{1}{\sin x}$	$\log_\epsilon \tan \dfrac{x}{2} + C$
$-\dfrac{\sin 2x}{\sin^4 x}$	$\dfrac{1}{\sin^2 x}$	$-\text{cotan} x + C$
$\dfrac{\sin^2 x - \cos^2 x}{\sin^2 x · \cos^2 x}$	$\dfrac{1}{\sin x · \cos x}$	$\log_\epsilon \tan x + C$
$n · \sin mx · \cos nx + m · \sin nx · \cos mx$	$\sin mx · \sin nx$	$\frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C$
$2a·\sin 2ax$	$\sin^2 ax$	$\dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C$
$-2a·\sin 2ax$	$\cos^2 ax$	$\dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C$

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