Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_0^\pi \sin ^2 x d x$ is equal to
A
$\pi$
✓
$\frac{\pi}{2}$
C
$0$
D
$\frac{\pi}{3}$
✓
Answer
Correct option: B.
$\frac{\pi}{2}$
(B) Let $I =\int_0^\pi \sin ^2 x d x=2 \int_0^{\pi / 2} \sin ^2 x d x$ $\ldots\left[\because \int_0^{2 a } f (x) d x=2 \int_0^{ a } f (x) d x\right.$, if $\left.f (2 a -x)= f (x)\right]$ If n is a positive integer, then $\int_0^{\pi / 2} \sin ^{n} x d x=\int_0^{\pi / 2} \cos ^{n} x d x$ $=\left\{\begin{array}{l}\frac{(n-1)(n-3) \ldots .2}{n(n-2) \ldots .3}, \text { when } n \text { is odd } \\ \frac{(n-1)(n-3) \ldots .1}{n(n-2) \ldots .2} \cdot \frac{\pi}{2}, \text { when } n \text { is even }\end{array}\right.$ $\therefore I=2 \times \frac{1}{2} \times \frac{\pi}{2}$ $=\frac{\pi}{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.