Q54 / 155 · WBJEE 2024 · Mathematics

Figure

  • A
    $\pi$
  • B
    1
  • Correct answer
    C
    0
  • D
    $\frac{3 \pi}{2}$

Solution

\(\begin{aligned} & I=\int_0^\pi e^{\cos ^2 x} \cos ^3(2 n+1) x d x, n \in Z \ldots(1) \\ & =\int_0^\pi e^{\cos ^2(\pi-x)} \cos ^3[(2 n+1)(\pi-x)] d x \end{aligned}\) Using \(\int_0^a f(x) d x=\int_0^a f(a-x) d x\) \(\begin{aligned} & \therefore I=\int_0^\pi e^{\cos ^2 x} \cos ^3[(2 n+1) \pi-(2 n+1) x] d x \\ & I=\int_0^\pi-e^{\cos ^2 x} \cos ^3(2 n+1) x d x \ldots \text { (2) } \end{aligned}\) Adding (1) and (2) we get \(21=0 \Rightarrow 1=0\)

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

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