MCQ
For any integer $n,$ the integral $\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $
- A$ - 1$
- ✓$0$
- C$1$
- D$\pi $
Since $\cos (2n + 1)(\pi - x) = \cos [(2n + 1)\pi - (2n + 1)x]$
$ = - \cos (2n + 1)x$ and ${\sin ^2}(\pi - x) = {\sin ^2}x$
Hence by the property of definite integral,
$\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = 0} $, $[f(2a - x) = - f(x)]$.
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$-3, -\frac{1}{3}$
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