Question

\[\int\limits_0^{\pi/2} \cos^2 x\ dx\]

Answer 1

\[Let I = \int_0^\frac{\pi}{2} \cos^2 x\ d\ x\ . Then, \]
\[I = \int_0^\frac{\pi}{2} \cos^2 x\ d\ x\]
\[ \Rightarrow I = \frac{1}{2} \int_0^\frac{\pi}{2} \left( 1 + \cos 2x \right) dx \left[ \because \cos 2x = 2 \cos^2 x - 1 \right]\]
\[ \Rightarrow I = \left[ \frac{x}{2} + \frac{\sin 2x}{4} \right]_0^\frac{\pi}{2} \]
\[ \Rightarrow I = \frac{\pi}{4} + 0 - 0\]
\[ \Rightarrow I = \frac{\pi}{4}\]