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Question: Answered & Verified by Expert
$$
\int_0^\pi \frac{x \cos ^2 x}{1+\sin x} d x=
$$
MathematicsDefinite IntegrationTS EAMCETTS EAMCET 2023 (13 May Shift 2)
Options:
  • A $\frac{\pi(\pi-2)}{2}$
  • B 1
  • C $\frac{\pi(x+2)}{2}$
  • D $\frac{\pi}{4}$
Solution:
1725 Upvotes Verified Answer
The correct answer is: $\frac{\pi(\pi-2)}{2}$
$\begin{aligned} & \text {} I=\int_0^\pi \frac{x \cos ^2 x}{1+\sin x} d x \\ & \Rightarrow I=\int_0^\pi \frac{(\pi-x) \cos ^2 x}{1+\sin x} d x \\ & \Rightarrow \quad 2 I=\int_0^\pi \frac{\pi \cos ^2 x}{1+\sin x} d x \\ & \Rightarrow 2 I=\pi \int_0^\pi \frac{1-\sin ^2 x}{1+\sin x} d x \\ & \Rightarrow I=\frac{\pi}{2} \int_0^\pi(1-\sin x) d x \\ & \Rightarrow I=\frac{\pi}{2}[x+\cos x]_0^\pi \\ & =\frac{\pi}{2}(\pi-1-1)=\frac{\pi(\pi-2)}{2}\end{aligned}$

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