Q118 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics
$\int_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x=$
- A$\frac{64 \sqrt{2}}{5}$
- B$\frac{128 \sqrt{2}}{5}$
- C$\frac{256 \sqrt{2}}{3}$
- Correct answerD$\frac{128 \sqrt{2}}{3}$
Solution
$\begin{aligned}
\quad \mathrm{I} & =\int_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x=2 \int_0^{5 \pi}\left(2 \sin ^2 x\right)^{\frac{5}{2}} d x \\
\Rightarrow \mathrm{I} & =(2)^3 \sqrt{2} \int_0^{5 \pi} \sin ^5 x d x
\end{aligned}$
Let $\frac{x}{10}=\theta \Rightarrow \mathrm{I}=10(2)^3 \sqrt{2} \int_0^{\frac{\pi}{2}} \sin ^5 10 \theta d \theta$ $I=10(2)^3 \sqrt{2} \times \frac{4 \times 2}{5 \times 3}=\frac{128 \sqrt{2}}{3}$
Let $\frac{x}{10}=\theta \Rightarrow \mathrm{I}=10(2)^3 \sqrt{2} \int_0^{\frac{\pi}{2}} \sin ^5 10 \theta d \theta$ $I=10(2)^3 \sqrt{2} \times \frac{4 \times 2}{5 \times 3}=\frac{128 \sqrt{2}}{3}$
Hence, the correct answer is option: 'D'.