| $\dfrac{dy}{dx}$ | $y$ | $\int y\, dx$ |
|---|
| 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 $ |