Q117 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

$\int_{-2}^2 x^4\left(4-x^2\right)^{\frac{7}{2}} d x=$

  • A
    $4 \pi$
  • B
    $\frac{\pi}{16}$
  • Correct answer
    C
    $28 \pi$
  • D
    $\frac{3 \pi}{128}$

Solution

Let $\mathrm{x}=2 \sin \theta, \mathrm{dx}=2 \cos \theta \mathrm{~d} \theta$ $\begin{aligned} & \mathrm{I}=\int_{-\pi / 2}^{\pi / 2} 2^4 \sin ^4 \theta\left(4-4 \sin ^2 \theta\right)^{7 / 2} \times 2 \cos \theta \mathrm{~d} \theta \\ & =2^{12} \times 2 \int_0^{\pi / 2} \sin ^4 \theta \cos ^8 \theta \mathrm{~d} \theta \\ & =\frac{2^{13} \times(4-1)(4-3)(8-1)(8-3)(8-5)(8-7)}{12 \times 10 \times 8 \times 6 \times 4 \times 2} \frac{\pi}{2}=28 \pi\end{aligned}$

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

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