Question

\[\int \left( a \tan x + b \cot x \right)^2 dx\]

Answer 1

\[\int \left( a \tan x + b \cot x \right)^2 dx\]
\[ = \int\left( a^2  \tan^2 x + b^2 \cot^2 x +\text{ 2ab tan x }\cot x \right)dx\]
\[ = a^2 \int \tan^2\text{ x  dx }+ b^2 \int \cot^2 \text{x dx }+ \text{2ab  ∫ dx}\] 
\[ = a^2 \int\left( \sec^2 x - 1 \right)dx + b^2 \int\left( {cosec}^2 x - 1 \right)dx + 2ab\  ∫ dx\]

\[ = a^2 \left[ \tan x - x \right] + b^2 \left[ - \cot x - x \right] + \text{2ab x }+ C\]
\[ = a^2 \tan x - b^2 \cot x - \left( a^2 + b^2 - 2ab \right)x + C\]