Question

\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]

Answer 1

\[Let\ I = \int_0^{( \pi )^\frac{2}{3}} \sqrt{x} \cos^2 x^\frac{3}{2} d x . Then, \]
\[Let\ x^\frac{3}{2} = t . Then, \frac{3}{2}\sqrt{x} dx = dt\]
\[When\ x = 0, t = 0\ and\ x = \left( \pi \right)^\frac{2}{3} , t = \pi\]
\[ \therefore I = \frac{2}{3} \int_0^\pi \cos^2 t dt\]
\[ \Rightarrow I = \frac{2}{3} \int_0^\pi \frac{1 + \cos 2x}{2} dx\]
\[ \Rightarrow I = \frac{1}{3} \left[ x + \frac{\sin 2x}{2} \right]_0^\pi \]
\[ \Rightarrow I = \frac{1}{3}\left( \pi + 0 \right)\]
\[ \Rightarrow I = \frac{\pi}{3}\]