Question

\[\int\limits_0^{\pi/4} \frac{\tan^3 x}{1 + \cos 2x} dx\]

Answer 1

\[Let\ I = \int_0^\frac{\pi}{4} \frac{\tan^3 x}{1 + \cos 2x} d\ x . Then, \]
\[I = \int_0^\frac{\pi}{4} \frac{\tan^3 x}{2 \cos^2 x} d\ x\]
\[ \Rightarrow I = \frac{1}{2} \int_0^\frac{\pi}{4} \tan^3 x \sec^2 x dx\]
\[Let \tan\ x = t . Then, \sec^2 x\ dx\ = dt\]
\[When\ x = 0, t = 0\ and\ x\ = \frac{\pi}{4}, t = 1\]
\[ \therefore I = \frac{1}{2} \int_0^1 t^3 dt\]
\[ \Rightarrow I = \frac{1}{2} \left[ \frac{t^4}{4} \right]_0^1 \]
\[ \Rightarrow I = \frac{1}{2}\left( \frac{1}{4} - 0 \right)\]
\[ \Rightarrow I = \frac{1}{8}\]