Q115 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

$\int_0^\pi x \sin ^4 x \cos ^6 x d x=$

  • Correct answer
    A
    $\frac{3 \pi^2}{512}$
  • B
    $\frac{3 \pi^2}{256}$
  • C
    $\frac{\pi^2}{256}$
  • D
    $\frac{\pi^2}{512}$

Solution

Let $I=\int_0^\pi x \sin ^4 x \cos ^6 x d x$ $\begin{aligned} & \Rightarrow I=\int_0^\pi(\pi-x) \sin ^4(\pi-x) \cos ^6(\pi-x) d x \\ & \Rightarrow I=\int_0^\pi(\pi-x) \sin ^4 x \cos ^6 x d x \\ & \Rightarrow 2I=2 \pi \int_0^{\pi / 2} \sin ^4 x \cos ^6 x d x \\ & \Rightarrow I=\pi\left(\frac{3 \cdot 1 \cdot 5 \cdot 3 \cdot 1}{10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \cdot \frac{\pi}{2}\right)=\frac{3 \pi^2}{512}\end{aligned}$

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

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