# I require a hint in solving this indefinite integral.

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
Here's what Wolfram Alpha thinks. Hopefully you can figure it out from there.

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.

Here's what Wolfram Alpha thinks. Hopefully you can figure it out from there.

#### greg1313

Forum Staff
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.

#### idontknow

$$\displaystyle I_n =uI_{n}' +(n+1)I_n +\int \dfrac{u^{-n}\left(nu^{n-1}-1\right)\left(u^n-u\right)^{\frac{1}{n}-1}}{n}du$$.
Try to simplify the remaining integral , knowing that $$\displaystyle u^{-n} (nu^{n-1} -1) =\dfrac{n-1}{u}+(\dfrac{1}{u}-\dfrac{1}{u^n })$$.

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$$\displaystyle I_n =I_{n}' +(n+1)I_n +\int \dfrac{u^{-n}\left(nu^{n-1}-1\right)\left(u^n-u\right)^{\frac{1}{n}-1}}{n}du$$.
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.

#### idontknow

$$\displaystyle I_n$$- is the integral , use integration by parts.
One correction : in the first line replace $$\displaystyle I_{n}'$$ with $$\displaystyle uI_{n}'$$.

$$\displaystyle \int \frac{(u^n -u)^{1/n}}{u^{n+1}}du$$ . No matter what substitution I make . . .
Divide the numerator and denominator by $u$, then substitute $v = u^{1 - n}$.