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Pattern – 15 : Definite integration

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⋆ Pattern – 15

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

$\int_{nT}^{a+nT}f\left(x\right)\ dx$ = $\int_{0}^{a}f\left(x\right)\ dx$ ; Where, n ∈ I & a ∈ R

Proof :

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

Let, x – nT = y

∴ x = y + nT

∴ dx = dy

x = nT ⇒ y = 0

x = a + nT ⇒ y = a

$\therefore\ \ I$ = $\ \int_{0}^{a}f\left(y+nT\right)dy$

$=\ \int_{0}^{a}f\left(y\right)dy$

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