Q116 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics
$\int_{-2}^2\left(4-x^2\right)^{\frac{5}{2}} d x=$
- A$40 \pi$
- Correct answerB$20 \pi$
- C$\frac{5 \pi}{32}$
- D$\frac{5 \pi}{32}$
Solution
$I=\int_{-2}^2\left(4-x^2\right)^{5 / 2} d x=2 \int_0^2\left(4-x^2\right)^{\frac{5}{2}} d x$
Let $x=2 \sin \theta \Rightarrow d x=2 \cos \theta d \theta$ $\begin{aligned} & I=2 \int_0^{\frac{\pi}{2}} 2 \cos \theta\left(4-4 \sin ^2 \theta\right)^{\frac{5}{2}} d \theta=128 \int_0^{\frac{\pi}{2}} \cos ^6 \theta d \theta \\ & \Rightarrow I=128 \times \frac{5 \times 3 \times 1}{6 \times 4 \times 2} \times \frac{\pi}{2}=20 \pi \end{aligned}$
Let $x=2 \sin \theta \Rightarrow d x=2 \cos \theta d \theta$ $\begin{aligned} & I=2 \int_0^{\frac{\pi}{2}} 2 \cos \theta\left(4-4 \sin ^2 \theta\right)^{\frac{5}{2}} d \theta=128 \int_0^{\frac{\pi}{2}} \cos ^6 \theta d \theta \\ & \Rightarrow I=128 \times \frac{5 \times 3 \times 1}{6 \times 4 \times 2} \times \frac{\pi}{2}=20 \pi \end{aligned}$
Hence, the correct answer is option: 'B'.