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Pattern – 21 : Definite integration

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⋆ Pattern – 21

If f(t) is an odd function, then

$\emptyset\left(x\right)\$ = $\ \int_{a}^{x}f\ \left(t\right)dt$ is an even function.

Proof :

Now,

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

Let, t = -y

∴ y = -t

∴ dt = -dy

t = a ⇒ y = -a

t = -x ⇒ y = x

Now,

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

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

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

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

= ∅ (x)

∴ ∅ is an even function