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December 28th, 2014, 11:47 AM  #1 
Newbie Joined: Nov 2014 From: kragujevac Posts: 8 Thanks: 0  recursive integration,integration done, how to get formula
Hi there . I need to integrate function dx/sin^n (x). I did the integration part, i believe it is right. In= In2 2/(n*sin^n(x) I am sorry for formula written like this , I am learning coding so I hope I'll soon be able to write it better. So now i thing the condition should be n>=2 and I started getting formula seperately for odd and even index but I can't see it ? 
December 28th, 2014, 12:40 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,214 Thanks: 2410 Math Focus: Mainly analysis and algebra 
$$\newcommand{\d}[1][x]{\,\mathrm d #1}\begin{align*} I_n &= \int \frac{\d}{\sin^n x} = \int \sin^{n} x \d \\ &= \int \left(1\cos^2 x\right) \sin^{n2} x \d \\ &= I_{n+2}  \int \underbrace{\cos x}_v \, \underbrace{\cos x \sin^{n2} x \d}_{\d[u]} & \text{by parts} \\ &= I_{n+2} + \cos x \, \frac{1}{n+1}\sin^{n1} x + \int \sin x \, \frac{1}{n+1}\sin^{n1} x \d \\ &= I_{n+2} + \frac{1}{n+1}\cos x \, \sin^{n1} x + \frac{1}{n+1}I_n \\ \frac{n+2}{n+1}I_n &= I_{n+2} + \frac{1}{n+1}\cos x \, \sin^{(n+1)} x \\ I_{n+2} &= \frac{n+2}{n+1}I_{n}  \frac{1}{n+1} \cos x \, \sin^{(n+1)}x \\[12pt] I_{n} &= \frac{n}{n1}I_{n2}  \frac{1}{n1} \cos x \, \sin^{(n1)}x \qquad n \gt 1 \end{align*}$$ It's possible that there are sign errors in there. There is no need for a separate formula for odd and even indices. We simply need to be able to evaluate $I_0$ and $I_1$ by other means (which we can). $$\begin{align*}I_1 &= \int \frac{\d}{\sin x} = \int \csc x \d \\ &= \int \frac{\csc x ( \csc x + \cot x )}{\csc x + \cot x} \d \\ &= \int \frac{\csc^2 x \csc x \cot x}{\csc x + \cot x} \d \\ &= \log {\left(A\left \csc x + \cot x \right\right)}\end{align*}$$ Last edited by v8archie; December 28th, 2014 at 01:00 PM. 

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formula, integration, recursive 
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