NIMCET 2018 — Mathematics PYQ

Concept:
- $\sin x$ is an odd function so, $\sin (-x)=-\sin x$
- $\cos x$ is an even function so, $\cos (-x)=\cos x$

Calculation:
Given, $f(x)={\sin }^{5}x+{\sin }^{3}x$ and $g(x)={\cos }^{6}x+{\sin }^{3}x$
$f(-x)=-{\sin }^{5}x-{\sin }^{3}x$ and $g(-x)={\cos }^{6}x-{\sin }^{3}x$
$[f(x)+f(-x)]={\sin }^{5}x+{\sin }^{3}x-{\sin }^{5}x-{\sin }^{3}x\Rightarrow [f(x)+f(-x)]=0$

$[g(x)+g(-x)]=2\times {\cos }^{6}x$

${\int }_0^{\pi /2}[f(x)+f(-x)][g(x)+g(-x)]dx$

$={\int }_0^{\pi /2}(0)\times (2{\cos }^{6}x)dx=0$