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Pattern – 17 : Definite integration

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⋆ Pattern – 17

If f is a periodic function with principal period T, then

$\int_{a}^{a+nT}f\left(x\right)\ dx$ = $n \int_{0}^{T}f\left(x\right)\ dx$

; Where, n ∈ I & a ∈ R

Proof :

Let, $\ I\ =\ \int_{a}^{a+nT}f\left(x\right)dx\$

$=\ \int_{a}^{0}f\left(x\right)dx+\ \int_{0}^{nT}f\left(x\right)dx+\ \int_{nT}^{a+nT}f\left(x\right)dx$

$=\ -\int_{0}^{a}f\left(x\right)dx+n\int_{0}^{T}f\left(x\right)dx+\ \int_{0}^{a}f\left(x\right)dx$

$=n\int_{0}^{T}f\left(x\right)dx$