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Pattern – 19 : Definite integration

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⋆ Pattern – 19 ( Odd Function )

If f is an odd periodic function with principal period T, then is an even periodic function with principal period T.

Proof :

Let, g(x) – $\int_{a}^{x}f\left(t\right)dt$ —–(1)

Now,

$g\left(x+T\right)=\ \int_{a}^{x+T}f\left(t\right)dt$

$=\ \int_{a}^{x}f\left(t\right)dt+\ \int_{x}^{x+T}f\left(t\right)dt$

$=g\left(x\right)+\ \int_{0}^{x}f\left(t\right)dt$

= g(x) + 0

= g(x)

∴ g has a p.p. T

Now,

$9\left(-x\right)$ = $\ \int_{a}^{-x}f\left(t\right)dt$

Let, t = -y

∴ -t = y

∴ dt = -dy

t = a ⇒ y = -a

t = -x ⇒ y = x

$\therefore\ \ 9\left(-x\right)$ = $\ -\int_{-a}^{x}f\left(-y\right)dy$

$=\ t\int_{-a}^{x}f\left(y\right)\ dy$

$=\ \int_{-a}^{a}f\left(y\right)dy+\int_{a}^{x}f\left(y\right)dy$

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

= 9(x) ∴ 9 is even