I require a hint in solving this indefinite integral.

Aug 2019
40
3
India
The problem I want to solve is this
\(\displaystyle \int\frac{ (\sin^n \theta - \sin\theta)^{1/n} \cos\theta}{\sin^{n+1}\theta} d\theta\)

Now, if I make a substitution of \(\displaystyle u = \sin\theta\) then, the integral would look like this

\(\displaystyle \int \frac{(u^n -u)^{1/n}}{u^{n+1}}du\) . No matter what substitution I make the problem is the thing inside of that \(\displaystyle 1/n \) power. So, can you please tell me how to write \(\displaystyle x^n -x\) in form of something else.
 

greg1313

Forum Staff
Oct 2008
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London, Ontario, Canada - The Forest City
Here's what Wolfram Alpha thinks. Hopefully you can figure it out from there.
 
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Aug 2019
40
3
India
T
Here's what Wolfram Alpha thinks. Hopefully you can figure it out from there.
Thank you. Can you give some real-life advice about how to for integrals, I mean something that you would have discovered during your time on indefinite integrals. Your experience will be better than any textbook’s corny explanation.
 

greg1313

Forum Staff
Oct 2008
8,008
1,174
London, Ontario, Canada - The Forest City
Practice , practice, practice...(as corny as it may sound).

I'm working on your integral but I've yet to come up with anything substantial. Maybe integration by parts...
Hopefully I'll be able to post something by tomorrow night.
 
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Dec 2015
972
128
Earth
\(\displaystyle I_n =uI_{n}' +(n+1)I_n +\int \dfrac{u^{-n}\left(nu^{n-1}-1\right)\left(u^n-u\right)^{\frac{1}{n}-1}}{n}du \).
Try to simplify the remaining integral , knowing that \(\displaystyle u^{-n} (nu^{n-1} -1) =\dfrac{n-1}{u}+(\dfrac{1}{u}-\dfrac{1}{u^n }) \).
 
Last edited:
Aug 2019
40
3
India
\(\displaystyle I_n =I_{n}' +(n+1)I_n +\int \dfrac{u^{-n}\left(nu^{n-1}-1\right)\left(u^n-u\right)^{\frac{1}{n}-1}}{n}du \).
Try to simplify the remaining integral , knowing that \(\displaystyle u^{-n} (nu^{n-1} -1) =\dfrac{n-1}{u}+(\dfrac{1}{u}-\dfrac{1}{u^n }) \).
I have really not understood, I request you to please write steps, can you please explain your very first line.
 
Dec 2015
972
128
Earth
\(\displaystyle I_n \)- is the integral , use integration by parts.
One correction : in the first line replace \(\displaystyle I_{n}' \) with \(\displaystyle uI_{n}'\).
 
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