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Definite integral

The fundamental theorem of calculus links differentiation and integration: to compute a definite integral we find an antiderivative FF and plug in the limits.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)the fundamental theorem of calculus, where F′=fF'=f

Symbols

FFan antiderivative of f
a, ba,\ blimits of integration

Example

∫0πsin⁡x dx=[−cos⁡x]0π=1−(−1)=2\displaystyle\int_0^{\pi}\sin x\,dx = [-\cos x]_0^{\pi} = 1-(-1) = 2.

Remember to substitute the upper limit minus the lower limit, in that order.
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Part of Calculus: Integration.