Integration

• Function => Integral

  $ax^n$ => $\frac{a}{n+1}x^{n+1}+$$C$
• Function => Integral

  $e^x$ => $e^x+$$C$
• Function => Integral

  $\frac{1}{x}$ => $\ln\left|x\right|+$$C$
• Function => Integral

  $\cos x$ => $\sin x\ +$$C$
• Function => Integral

  $\sin x$ => $-\cos x+$$C$
• Function => Integral

  $\sec x\tan x$ => $\sec x+$$C$
• Function => Integral

  $\operatorname{cosec}^2x$ => $-\cot x+$$C$
• Function => Integral

  $f'\left(ax+b\right)$ => $\frac{1}{a}f\left(ax+b\right)+$$C$
• Integration by parts

  $\int_{ }^{ }uv'dx=$ $uv-\int_{ }^{ }u'vdx$
• Parametric integration

  $\int_{ }^{ }y\frac{dx}{dt}dt$

  and remember to change the limits
• To help with integration you can split the numerator

  $\frac{x-1}{x}=$ $\frac{x}{x}-\frac{1}{x}$
• To help with integration you can use the reverse chain rule

  $\int_{ }^{ }\frac{du}{dx}f'\left(u\right)dx=$ $f\left(u\right)+$$c$
• To help with integration you can use algebraic division
• To help with integration you can use partial fractions
• Trapezium rule

  $\int_{ }^{ }ydx\approx$ $\frac{1}{2}h\left(F+L+2M\right)$

  where F=first, L=last and M=middle