Q114 / 160 · AP EAMCET 2023 (19 May Shift 1) · Mathematics

If $a=2 n$ and $b=2 m+1$ for all $m, n \in \mathbb{N}$, $$ \int_{-\pi}^\pi e^{\sin ^a x} \cot ^b(2 n+1) x d x= $$

  • Correct answer
    A
    $0$
  • B
    $1$
  • C
    $-1$
  • D
    $\pi$

Solution

Given $a=2 n, b=2 m+1$ and $m, n \in \mathrm{N}$ Let $I=\int_{-\pi}^\pi e^{\sin ^a x} \cot ^b(2 n+1) x d x$ Let $f(x)=e^{\sin ^a x} \cdot \cot ^b(2 n+1) x$ $$ \Rightarrow \quad f(-x)=e^{\sin ^a(-x)} \cot ^b(2 n+1)(-x) $$ Since $a$ is even number and $b$ is odd number and $\cot (-\theta)=-\cot \theta$ $$ \begin{aligned} & \Rightarrow f(-x)=e^{\sin ^a(x)} \cdot\left[-\cot ^b(2 n+1) x\right] d x \\ & \Rightarrow f(-x)=-f(x) \end{aligned} $$ Hence $$ I=\int_{-\pi}^\pi f(x) d x=0\left\{\begin{array}{l} \because f(-x)=-f(x) \\ \int_{-a}^a f(x) d x=0 \end{array}\right. $$

Hence, the correct answer is option: 'A'.

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