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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=- In-2 -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: 7,620 Thanks: 2609 Math Focus: Mainly analysis and algebra \newcommand{\d}[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^{-n-2} x \d \\ &= I_{n+2} - \int \underbrace{\cos x}_v \, \underbrace{\cos x \sin^{-n-2} x \d}_{\d[u]} & \text{by parts} \\ &= I_{n+2} + \cos x \, \frac{1}{n+1}\sin^{-n-1} x + \int \sin x \, \frac{1}{n+1}\sin^{-n-1} x \d \\ &= I_{n+2} + \frac{1}{n+1}\cos x \, \sin^{-n-1} 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}{n-1}I_{n-2} - \frac{1}{n-1} \cos x \, \sin^{-(n-1)}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. Tags formula, integration, recursive Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post bongantedd Algebra 7 September 21st, 2013 11:43 PM Joe_65 Calculus 2 January 23rd, 2013 09:43 PM nappysnake Calculus 6 November 30th, 2011 05:29 AM zg12 Calculus 1 March 6th, 2010 10:57 AM brunojo New Users 1 November 30th, 2007 11:59 AM

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