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Table of Integrals

Basic Forms

  • {$ \large \int 0 dx = C $}
  • {$ \large \int k dx = kx $}
  • {$ \large \int k f(x)dx = k \int f(x) dx $}
  • {$ \large \int x^n dx = {{x^{n+1}} \over {n+1}} , n \ne -1 $}
  • {$ \large \int {{dx}\over x} = \ln |x| $}

Techniques

  • {$ \large \int uv' dx = uv - \int u'v dx \; \text{ (integration by parts)} $}
  • {$ \large \int \{f(x) + g(x)\}dx = \int f(x) dx + \int g(x)dx $}
  • {$ \large \int F\{f(x)\} dx= \int F(u){{dx} \over {du}}du = \int {{F(u)} \over {f^\prime(x)}}du $}

Rational Functions

  • {$ \large \int {dx \over \sqrt{1 - x^2}} = \arcsin x $} (Demo)
  • {$ \large \int {dx \over {1 + x^2}} = \arctan x $}

Exponential Functions

  • {$ \large \int f^\prime(x)e^{f(x)} dx = e^{f(x)} $}
  • {$ \large \int e^{cx} dx = \frac{1}{c}e^{cx} $}
  • {$ \large \int xe^x dx = xe^x - e^x $}
  • {$ \large \int x e^{x^2} dx = \frac{1}{2} e^{x^2} $}

Logarithmic Functions

Trigonometric Functions

  • {$ \large \int \cos x dx = \sin x $}
  • {$ \large \int \sin x dx = -\cos x $}
  • {$ \large \int \tan x dx = -log \cos x $}
  • {$ \large \int \omega\cos\omega t dt = \sin \omega t $}
  • {$ \large \int \omega\sin\omega t dt = -\cos \omega t $}
  • {$ \large\int t \sin t dt = -t\cos t + \sin t $} (Demo)
  • {$ \large \int 2\sin x \cos x dx = \sin^2 x $}
  • {$ \large \int \cos\theta\log \sin \theta d\theta = \sin\theta( \log \sin \theta - 1) $} (Demo)
  • {$ \large \int \tan^2 dx = \tan x - x $}
  • {$ \large \int \sec^4\theta\ d\theta = \frac{1}{3}\tan^3\theta + \tan\theta $} (Demo)

Inverse Trigonometric Functions

  • {$ \large\int \arcsin\ x dx = x\ \arcsin\ x + \sqrt{1 - x^2} $} (Demo)
  • {$ \large \int x^2\ \arcsin\ x\ dx = \frac{1}{3}x^3 \arcsin x + \frac{1}{9}x^2\sqrt{1-x^2} + \frac{2}{9}\sqrt{1-x^2} $} (Demo)

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Categories: TableOfIntegrals


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