Q116 / 160 · TS EAMCET 2023 (13 May Shift 1) · Mathematics

$\int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x=$

  • A
    $\frac{5 \pi}{16}$
  • B
    $\frac{5}{4}$
  • Correct answer
    C
    $\frac{5 \pi}{8}$
  • D
    $\frac{5}{8}$

Solution

$\int_0^2 x^{5 / 2} \sqrt{2-x} d x$ Put $x=2 \sin ^2 \theta, d x=4 \sin \theta \cos \theta d \theta$ $$ \begin{aligned} & =\int_0^{\pi / 2} x^{5 / 2} \sqrt{2-x} d x \\ & =\int_0^{\pi / 2}\left(2 \sin ^2 \theta\right)^{5 / 2} \sqrt{2-2 \sin ^2 \theta} \cdot 4 \sin \theta \cos \theta d \theta \\ & =\int_0^{\pi / 2} 2^3 \times 4 \cos ^2 \theta \sin ^6 \theta d \theta=32 \int_0^{\pi / 2} \sin ^6 \theta \cos ^2 \theta d \theta \end{aligned} $$ Using Walli's formula $$ =32\left[\frac{(5 \cdot 3 \ldots 1)(1)}{8 \cdot 6 \cdot 4 \cdot 2}\right] \cdot \frac{\pi}{2}=\frac{5 \pi}{8} $$

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

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