Question

If n is any integer, then ${\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx=$ [IIT 1985; RPET 1995; UPSEAT 2001]

• A. x
• B. 1
• C. 0
• D. None of these

Answer 1

The correct option is C

0

Since $cos(2n+1)(\pi -x)=cos\left[\right(2n+1)\pi -(2n+1\left)x\right]$
$=-cos(2n+1)x$ and $co{s}^2(\pi -x)=co{s}^{2}x$
So that $f(2a-x)=-f\left(x\right)$, and hence by the property of definite integral
${\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx=0$