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.
$-\dfrac{1}{(x + a)^2}$ $\dfrac{1}{x + a}$ $\log_\epsilon (x+a) + C$
$-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}$ $\dfrac{1}{\sqrt{a^2 + x^2}}$ $\log_\epsilon (x + \sqrt{a^2 + x^2}) + C$
$\mp \dfrac{b}{(a ± bx)^2}$ $\dfrac{1}{a ± bx}$ $± \dfrac{1}{b} \log_\epsilon (a ± bx) + C$
$-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}$ $\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}$ $\dfrac{x}{\sqrt{a^2 + x^2}} + C$
$a · \cos ax$ $\sin ax$ $-\dfrac{1}{a} \cos ax + C$
$-a · \sin ax$ $\cos ax$ $\dfrac{1}{a} \sin ax + C$
$a · \sec^2ax$ $\tan ax$ $-\dfrac{1}{a} \log_\epsilon \cos ax + C$
$\sin 2x$ $\sin^2 x$ $\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$
$-\sin 2x$ $\cos^2 x$ $\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$
$n · \sin^{n-1} x · \cos x$ $\sin^n x$ $-\frac{\cos x}{n} \sin^{n-1} x + \frac{n-1}{n} \int \sin^{n-2} x\, dx + C$
$-\dfrac{\cos x}{\sin^2 x}$ $\dfrac{1}{\sin x}$ $\log_\epsilon \tan \dfrac{x}{2} + C$
$-\dfrac{\sin 2x}{\sin^4 x}$ $\dfrac{1}{\sin^2 x}$ $-\text{cotan} x + C$
$\dfrac{\sin^2 x - \cos^2 x}{\sin^2 x · \cos^2 x}$ $\dfrac{1}{\sin x · \cos x}$ $\log_\epsilon \tan x + C$
$n · \sin mx · \cos nx + m · \sin nx · \cos mx$ $\sin mx · \sin nx$ $\frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C$
$2a·\sin 2ax$ $\sin^2 ax$ $\dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C$
$-2a·\sin 2ax$ $\cos^2 ax$ $\dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C$

Main Page ↑