Question

For any integer n the integral ${\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx$ has the value

• A. $\pi$
• B. $1$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

$I={\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx,nϵZ...\left(1\right)={\int }_0^{\pi }{e}^{co{s}^2(\pi -x)}co{s}^3\left[\right(2n+1\left)\right(\pi -x\left)\right]dxUsing{\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx\therefore I={\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3\left[\right(2n+1)\pi -(2n+1\left)x\right]dxI={\int }_0^{\pi }-e^{co{s}^{2}x}co{s}^3(2n+1)xdx...\left(2\right)$
Adding (1) and (2) we get
$2I=0\Rightarrow I=0$