Table of Standard Forms

 

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(1\)\(x\)\(\frac{1}{2} x^2 + C\)
\(0\)\(a\)\(ax + C\)
\(1\)\(x \pm a\)\(\frac{1}{2} x^2 \pm ax + C\)
\(a\)\(ax\)\(\frac{1}{2} ax^2 + C\)
\(2x\)\(x^2\)\(\frac{1}{3} x^3 + C\)
\(nx^{n-1}\)\(x^n\)\(\dfrac{1}{n+1} x^{n+1} + C\)
\(-x^{-2}\)\(x^{-1}\)\(\ln |x| + C\)
\(\dfrac{du}{dx} \pm \dfrac{dv}{dx} \pm \dfrac{dw}{dx}\)\(u \pm v \pm w\)\(\int u\, dx \pm \int v\, dx \pm \int w\, dx\)
\(u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}\)\(uv\)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:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(e^x\)\(e^x\)\(e^x + C\)
\(x^{-1}\)\(\ln x\)\(x(\ln x - 1) + C\)
\(\dfrac{1}{x\,\ln b}\)\(\log_{b} x\)\(\dfrac{1}{\ln b} x (\ln x - 1) + C\)
\(a^x \ln a\)\(a^x\)\(\dfrac{a^x}{\ln a} + C\)

Trigonometrical:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(\cos x\)\(\sin x\)\(-\cos x + C\)
\(-\sin x\)\(\cos x\)\(\sin x + C\)
\(\sec^2 x\)\(\tan x\)\(-\ln|\cos x| + C\)
\(\sec x\ \tan x\)\(\sec x\)\(\ln|\sec x+\tan x| + C\)

Inverse Trigonometrical:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(\dfrac{1}{\sqrt{1-x^2}}\)\(\arcsin x=\sin^{-1} x\)\(x \cdot \arcsin x + \sqrt{1 - x^2} + C\)
\(-\dfrac{1}{\sqrt{1-x^2}}\)\(\arccos x=\cos^{-1} x\)\(x \cdot \arccos x - \sqrt{1 - x^2} + C\)
\(\dfrac{1}{1+x^2}\)\(\arctan x=\tan^{-1} x\)\(x \cdot \arctan x - \frac{1}{2} \ln (1 + x^2) + C\)

Hyperbolic:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(\cosh x\)\(\sinh x\)\(\cosh x + C\)
\(\sinh x\)\(\cosh x\)\(\sinh x + C\)
\(\text{sech}^2 x\)\(\tanh x\)\(\ln \cosh x + C\)
\(-\text{sech } x \tanh x\)\(\text{sech }x\)\(2\arctan\left(e^x\right)+C\)

Inverse Hyperbolic:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(\dfrac{1}{\sqrt{x^2+1}}\)\(\text{arcsinh } x=\sinh^{-1} x\)\(-\sqrt{1 + x^2} + x \text{ arcsinh }x+C\)
\(\dfrac{1}{\sqrt{x^2-1}}\)\(\text{arccosh } x=\cosh^{-1} x\)\(-\sqrt{x^2-1}+x \text{ arccosh }x+C\)
\(\dfrac{1}{1-x^2}\)\(\text{arctanh }x = \tanh^{-1} x\)\(x \text{ arctanh } x + \dfrac{1}{2} \ln|1 - x^2|+C\)

 

Miscellaneous:

\(\dfrac{dy}{dx}\)\(\longleftarrow\quad y\quad\longrightarrow\)\(\int y\, dx\)
\(-\dfrac{1}{(x + a)^2}\)\(\dfrac{1}{x + a}\)\(\ln |x+a| + C\)
\(\dfrac{x}{\sqrt{x^2+a^2}}\)\(\sqrt{a^2+x^2}\)\(\dfrac{1}{2}x\sqrt{x^2+a^2}+\dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C\)
\(-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}\)\(\dfrac{1}{\sqrt{a^2 + x^2}}\)\(\ln (x + \sqrt{a^2 + x^2}) + C\)
\(\mp \dfrac{b}{(a \pm bx)^2}\)\(\dfrac{1}{a \pm bx}\)\(\pm \dfrac{1}{b} \ln |a \pm bx|+ C\)
\(-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}\)\(\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}\)\(\dfrac{x}{\sqrt{a^2 + x^2}} + C\)
\(a \cdot \cos ax\)\(\sin ax\)\(-\dfrac{1}{a} \cos ax + C\)
\(-a \cdot \sin ax\)\(\cos ax\)\(\dfrac{1}{a} \sin ax + C\)
\(a \cdot \sec^2ax\)\(\tan ax\)\(-\dfrac{1}{a} \ln |\cos ax| + C\)
\(\sin 2x\)\(\sin^2 x\)\(\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C\)
\(-\sin 2x\)\(\cos^2 x\)\(\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C\)
\(n \cdot \sin^{n-1} x \cdot \cos x\)\(\sin^n x\)\(-\dfrac{\cos x}{n} \sin^{n-1} x + \dfrac{n-1}{n} \int \sin^{n-2} x\, dx\)
\(-\dfrac{\cos x}{\sin^2 x}\)\(\dfrac{1}{\sin x}\)\(\ln\left|\tan \dfrac{x}{2}\right| + C\)
\(-\dfrac{\sin 2x}{\sin^4 x}\)\(\dfrac{1}{\sin^2 x}\)\(-\cot x + C\)
\(\dfrac{\sin^2 x - \cos^2 x}{\sin^2 x \cdot \cos^2 x}\)\(\dfrac{1}{\sin x \cdot \cos x}\)\(\ln|\tan x| + C\)
\(n \cdot \sin mx \cdot \cos nx + m \cdot \sin nx \cdot \cos mx\)\(\sin mx \cdot \sin nx\)\(\frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C\)
\(2a\cdot\sin 2ax\)\(\sin^2 ax\)\(\dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C\)
\(-2a\cdot\sin 2ax\)\(\cos^2 ax\)\(\dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C\)