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For simplicy, I’ve excluded

  • \(dx\) for pre-integration
  • \(+ c\) for post-integration
Pre-Integration Post-Integration
Basic \(x^n, n \ne -1\) \(\frac{x^{n+1}}{n+1}\)
\(\frac{1}{x}\) \(\log x\)
\(e^x\) \(e^x\)
\(a^x\) \(\frac{a^x}{\log a}\)
Coefficient \(f(ax+b)\) \(\frac{F(ax + b)}{a}\)
Trignometric \(\sin x\) \(- \cos x\)
\(\cos x\) \(\sin x\)
\(\tan x\) \(\log \vert \sec x\vert\) \(-\log\vert \cos x \vert\)
\(\cot x\) \(\log \vert \sin x\vert\) \(-\log\vert \text{cosec } x \vert\)
\(\sec x\) \(\log\vert \sec x + \tan x\vert\) \(-\log\vert \sec x - \tan x \vert\)
\(\text{cosec }x\) \(\log\vert \text{cosec } x - \cot x\vert\) \(-\log\vert \text{cosec } x + \cot x \vert\)
\(\sec x \tan x\) \(\sec x\)
\(\text{cosec }x \cot x\) \(-\text{cosec } x\)
\(\sec^2 x\) \(\tan x\)
\(\text{cosec}^2 x\) \(- \cot x\)
IDK \(\frac{1}{\sqrt{1-x^2}}\) \(\sin^{-1} x\) \(-\cos^{-1} x\)
\(\frac{1}{\sqrt{1+x^2}}\) \(\tan^{-1} x\) \(-\cot^{-1} x\)
\(\frac{1}{x \sqrt{x^2 - 1}}\) \(\sec^{-1} x\) \(- \text{ cosec}^{-1} x\)
Squares \(\frac{1}{a^2 + x^2}\) \(\frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right)\)
\(\frac{1}{x^2 - a^2}\) \(\frac{1}{2a} \log\left\vert \frac{x-a}{x+a}\right \vert\)
\(\frac{1}{a^2 - x^2}\) \(\frac{1}{2a} \log\left\vert \frac{a+x}{a-x}\right \vert\)
Den Roots \(\frac{1}{\sqrt{a^2 - x^2}}\) \(\sin^{-1} \left( \frac{x}{a} \right)\)
\(\frac{1}{\sqrt{x^2 + a^2}}\) \(\log\left\vert x + \sqrt{x^2 + a^2} \right \vert\)
\(\frac{1}{x \sqrt{x^2 - a^2}}\) \(\frac{1}{a} \sec^{-1} \left(\frac{x}{a}\right)\)
Num Roots \(\sqrt{a^2 - x^2}\) \(\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right)\)
\(\sqrt{a^2 + x^2}\) \(\frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \log \vert x + \sqrt{a^2 + x^2} \vert\)
\(\sqrt{x^2 - a^2}\) \(\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log \vert x + \sqrt{x^2 - a^2} \vert\)
IDK \(e^x \Big(f(x) + f'(x) \Big)\) \(e^x f(x)\)
\(x \Big(f(x) + f'(x) \Big)\) \(x f(x)\)
Parts/ILATE \(\int (uv) dx\) \(u \int vdx + \int \left(u' {\small \int} vdx \right)\)

Partial Fractions

Function Partial Fraction
\(\frac{px+q}{(x-a)(x-b)}\) \(\frac{A}{(x-a)} + \frac{B}{(x-b)}\)
\(\frac{px^2 + qx + r}{(x-a)(x-b)(x-c)}\) \(\frac{A}{(x-a)} + \frac{B}{(x-b)} + \frac{C}{(x-c)}\)
\(\frac{px+q}{(x-a)^3}\) \(\frac{A}{(x-a)} + \frac{B}{(x-a)^2} + \frac{C}{(x-a)^3}\)
\(\frac{px^2 + qx + r}{(x-a)^2 (x-b)}\) \(\frac{A}{(x-a)} + \frac{B}{(x-a)^2} + \frac{C}{(x-b)}\)
\(\frac{px^2 + qx + r}{(x-a) (x^2 + bx + c)}\) \(\frac{A}{(x-a)} + \frac{Bx + C}{(x^2 + bx + c)}\)

Properties
\(\left(\int f(x) \cdot dx \right)'\) \(f(x)\)
\(\int f'(x) dx\) \(f(x) + c\)
\(\int k \cdot f(x) dx\) \(k \int f(x) dx\)
\(\int \Big(f(x) \pm g(x) \Big) \ dx\) \(\int f(x) \ dx \pm \int g(x) \ dx\)
Last Updated: 2024-01-24 ; Contributors: AhmedThahir

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