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Pattern – 16 : Definite integration

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⋆ Pattern – 16

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

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

; Where, n ∈ I & a ∈ R

Proof :

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

Let, x – nT = y

∴ x = y + nT

∴ dx = dy

x = a + nT ⇒ y = a

x = b + nT ⇒ y = b

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

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

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