How to tackle the integral ${\int}_{0}^{\pi}\frac{3\pi {x}^{2}-2{x}^{3}}{(1+\mathrm{sin}x{)}^{n}}dx,\text{where}n\in N$

nidantasnu
2022-07-02
Answered

How to tackle the integral ${\int}_{0}^{\pi}\frac{3\pi {x}^{2}-2{x}^{3}}{(1+\mathrm{sin}x{)}^{n}}dx,\text{where}n\in N$

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cefflid6y

Answered 2022-07-03
Author has **13** answers

Let $t=\mathrm{tan}y$

$\begin{array}{rl}{J}_{n}& ={\int}_{0}^{\frac{\pi}{4}}{\mathrm{sec}}^{2n}y\phantom{\rule{mediummathspace}{0ex}}dy={\int}_{0}^{1}({t}^{2}+1{)}^{n-1}dt=\sum _{k=0}^{n-1}\frac{{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{k}{\textstyle )}}{2(n-k)-1}\end{array}$

$\begin{array}{rl}{J}_{n}& ={\int}_{0}^{\frac{\pi}{4}}{\mathrm{sec}}^{2n}y\phantom{\rule{mediummathspace}{0ex}}dy={\int}_{0}^{1}({t}^{2}+1{)}^{n-1}dt=\sum _{k=0}^{n-1}\frac{{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{k}{\textstyle )}}{2(n-k)-1}\end{array}$

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