Question

\[\int \tan^{3/2} x \sec^2 \text{x dx}\]

Answer 1

\[\int \tan^\frac{3}{2} x \cdot \sec^2 \text{x dx}\]
\[Let \tan x = t\]
\[ \Rightarrow \sec^2 x = \frac{dt}{dx}\]
\[ \Rightarrow \sec^2 \text{x dx} = dt\]
\[Now, \int \tan^\frac{3}{2} x \cdot \sec^2 \text{x dx} \]
\[ = \int t^\frac{3}{2} dt\]

` = [t^{3/2 +1}/{3/2+1}]  + C`

\[ = \frac{2}{5} t^\frac{5}{2} + C\]
\[ = \frac{2}{5} \tan^\frac{5}{2} x + C\]