Question

\[\int {cosec}^4 2x\ dx\]

Answer 1

\[\text{ Let I } = \int {cosec}^4 \text{ 2x dx}\]

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

\[ = \int\left( 1 + \cot^2 2x \right) \cdot {cosec}^2 \text{ 2x  dx }\]

\[\text{ Putting   cot 2x = t}\]

\[ \Rightarrow - {cosec}^2 \left( 2x \right) \cdot \text{  2 dx = dt}\]

\[ \Rightarrow {cosec}^2 \left( 2x \right) \cdot dx = \frac{- dt}{2}\]

\[ \therefore I = - \frac{1}{2}\int\left( 1 + t^2 \right) \cdot dt\]

\[ = - \frac{1}{2} \left[ t + \frac{t^3}{3} \right] + C\]

\[ = - \frac{1}{2}\cot 2x + \frac{1}{6} \cot^3 2x + C ...........\left[ \because t = \cot 2x \right]\]