Question

\[\int \text{cosec}^2 x \text{ cos}^2 \text{  2x  dx} \]

Answer 1

\[\int \text{cosec}^2 x  .\text{ cos}^2 \text{  2x  dx} \]
\[ \Rightarrow \int\text{ cosec}^2 x \left( 1 - 2 \sin^2 x \right)^2 dx\]
\[ \Rightarrow \int \text{ cosec}^2 x \left( 1 + 4 \sin^4 x - 4 \sin^2 x \right)dx\]
\[ \Rightarrow \int\left( {cosec}^2 x + 4 \sin^2 x - 4 \right)dx\]
\[ \Rightarrow \int {cosec}^2 x \text{ dx} + 4\int\left( \frac{1 - \cos 2x}{2} \right)dx - 4\int dx\]
\[ \Rightarrow - \cot x + 2 \left[ x - \frac{\sin 2x}{2} \right] - 4x + C\]
\[ \Rightarrow - \cot x + 2x - \sin 2x - 4x + C\]
\[ \Rightarrow - \cot x - \sin 2x - 2x + C\]