_ - ,a ^a ,f ,(x) ,dx = - Vidyadip

MCQ

$\int\limits_{ - \,a}^a {\,f\,(x)\,dx} $=

✓
$\int\limits_0^a {\,\left[ {f\,(x)\,\, + \,\,f\,( - \,x)} \right]\,dx} $
B
$\int\limits_0^a {\,\left[ {f\,(x)\,\, - \,\,f\,( - \,x)} \right]\,dx} $
C
$2$ $\int\limits_0^a {\,f\,(x)\,dx} $
D
$Zero$

✓

Answer

Correct option: A.

$\int\limits_0^a {\,\left[ {f\,(x)\,\, + \,\,f\,( - \,x)} \right]\,dx} $

a
$I =$ $\int\limits_{ - \,a}^a {\,f\,(x)\,dx} $ =$\int\limits_{ - a}^a {f( - x)\,dx} $ (using $K$)
$\therefore$ $2I$ = $\int\limits_{ - \,a}^a {\,\left( {f\,(x) + f( - x)} \right)\,dx} $ =$2\,\int\limits_0^a {\,\left( {f\,(x) + f( - x)} \right)\,dx} $ (as integral is even)

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