Question

\[\int \tan^3 x\ dx\]

Answer 1

\[\text{ Let I } = \int \tan^3 x \text{ dx }\]
\[ = \int\tan x \cdot \tan^2 x\text{  dx }\]
\[ = \int\tan x \left( \sec^2 x - 1 \right)dx\]
\[ = \int\tan x \cdot \sec^2 x \text{ dx} - \int\text{ tan x dx }\]
\[\text{ Putting   tan x }= t\ in\ the\ Ist\ integral\]
\[ \Rightarrow \text{ sec}^2 \text{ x dx }= dt\]
\[ \therefore I = \int t \cdot dt - \int\text{ tan  x  dx }\]
\[ = \frac{t^2}{2} - \text{ ln }\left| \sec x \right| + C\]
\[ = \frac{\tan^2 x}{2} - \text{ ln }\left| \sec x \right| + C .............\left[ \because t = \tan x \right]\]