If $f(x)\ and\ g(x),\ x\epsilon R$ are continuous functions, then value of integral $\int_{\frac{- \pi}{2}}^{\frac{\pi}{2}}\left\lbrack \left\{ f(x) + f( - x) \right\}\left\{ g(x) - g( - x) \right\} \right\rbrack\ dx$ is

Question

Dr. Abubacker · Accepted Answer

(d)
Let $I = \int_{\frac{- \pi}{2}}^{\frac{\pi}{2}}{\left\{ f(x) + f( - x) ight\}\left\{ g(x) - g( - x) ight\}}\ dx$
Again, let
$h(x) = \left\{ f(x) + f( - x) ight\}\{ g(x) - g( - x)\}$
$\Rightarrow h( - x) = \left\{ f( - x) + f(x) ight\}\{ g( - x) - g(x)\}$
$\Rightarrow h( - x) = - h(x)$
Hence, $h(x)$ is an odd function
$	herefore\ I = 0$

Q. If $f(x)\ and\ g(x),\ x\epsilon R$ are continuous functions, then value of integral $\int_{\frac{- \pi}{2}}^{\frac{\pi}{2}}\left\lbrack \left\{ f(x) + f( - x) \right\}\left\{ g(x) - g( - x) \right\} \right\rbrack\ dx$ is

Explanation

Reviewed by Dr. Abubacker

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