Evaluate the following integrals.
$\int {e}^{x}{\mathrm{cot}}^{3}{e}^{x}dx$

Answer & Explanation

Nathanael Webber

Skilled2021-10-20Added 117 answers

Step 1
To solve the given integral
Step 2
given that
$=\int {e}^{x}{\mathrm{cot}}^{3}{e}^{x}dx$
put
${e}^{x}=u$ ${e}^{x}dx=du$ $=\int {\mathrm{cot}}^{3}udu$
using integral reduction
$=-\frac{{\mathrm{cot}}^{2}u}{2}-\int \mathrm{cot}udu$ $=-\frac{{\mathrm{cot}}^{2}u}{2}-\mathrm{ln}\left|\mathrm{sin}u\right|$ $=-\frac{{\mathrm{cot}}^{2}{e}^{x}}{2}-\mathrm{ln}\mid {\mathrm{sin}e}^{x}\mid +c$