Antiderivative

Tags

Calculus

Cegep/2

Word count

215 words

Reading time

2 minutes

The function $F(x)$ is an antiderivative of $f(x)$ if $F^{\prime }=f$.

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Antiderivatives are not unique. $F(x)+c$ is the general antiderivative of $f(x)$.
If $g(x)$ and $h(x)$ are both antiderivatives of $f(x)$, then $g(x)=h(x)+c$.

Evaluation

Tools:

• Rationalisation
• Factorisation
• Long division
• Trigonometric identities
  • Generally used before substitution
  • ==${\cos }^{2}x=\frac{1}{2}(1+\cos 2x)$==
  • ==${\sin }^{2}x=\frac{1}{2}(1-\cos 2x)$==
  • ${\sec }^{2}x={\tan }^{2}x+1$
  • ${\tan }^{2}x={\sec }^{2}x-1$
  • $\sin (2x)=2\sin x\cos x$
• Indefinite integrals
• u-substitution
• Trigonometric substitution
• Integration by parts

For trigonometric integrals:

• $\int {\sin }^{n}x{\cos }^{m}x\mathrm{d}x$
  • Easy: at least one is odd
  • Hard: both are even => ${\sin }^{2}x=\frac{1}{2}(1-\cos (2x))$
• $\int {\sec }^{n}x{\tan }^{m}x\mathrm{d}x$
  • Easy: odd $\tan x$, even $\sec x$
  • Hard:
    • $\int {\tan }^{2}x\mathrm{d}x⟹\int ({\sec }^{2}x-1)\mathrm{d}x$
    • $\int {\tan }^{3}x\mathrm{d}x⟹\int ({\sec }^{2}x-1)\tan x\mathrm{d}x$
    • $\int {\sec }^{3}x{\tan }^{2}x\mathrm{d}x⟹\int {\sec }^{3}x({\sec }^{2}x-1)\mathrm{d}x$
    • $\int {\sec }^{3}x\mathrm{d}x$ => IBP boomerang

Contributors

Ajitani Hifumi

Changelog

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