Question

If $\int cose{c}^{n}xdx=\frac{-cose{c}^{n-2}xcotx}{n-1}+A\int cose{c}^{n-2}xdx$ then A is equal to

• A. $\frac{1}{n-2}$
• B. $\frac{n}{n-2}$
• C. $\frac{n-1}{n-2}$
• D. $\frac{n-2}{n-1}$

Answer 1

The correct option is D $\frac{n-2}{n-1}$

$I_n=\int {\text{cosec}}^{n}x=\int {\text{cosec}}^{n-2}x{\text{cosec}}^{2}xdxu=cotxdu=-{\text{cosec}}^{2}xdx{I}_n=-\int {\text{cosec}}^{n-2}xd\left(cotx\right)I_n=-{\text{cosec}}^{n-2}xcotx+\int \left({\cot }^{2}x\right)(n-2){\text{cosec}}^{n-2}xdx{I}_n=-{\text{cosec}}^{n-2}xcotx+\int ({\text{cosec}}^{2}x-1)(n-2){\text{cosec}}^{n-2}xdx{I}_n=-{\text{cosec}}^{n-2}xcotx-(n-2)I_n+(n-2)I_{n-2}{I}_n=\frac{-{\text{cosec}}^{n-2}xcotx}{n-1}+\frac{n-2}{n-1}{I}_{n-2}$