# Weird Integral

## Homework Statement

An Integral : $$\int \frac{1}{x^n(1+x^n)^{1/n}} \;\mathrm{d}x}$$

## Homework Equations

The Standard integrals.

## The Attempt at a Solution

I'm aware that integrals like this become very easy after a clever substitution...but maybe I'm not that clever so I can't even start it. If anyone shows me the first step I'll try to take it from there.

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ideasrule
Homework Helper

$$\int \frac{1}{x^n(1+x^n)^{1/n}} \mathrm{d}x$$

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$$\int \frac{1}{x^n(1+x^n)^{1/n} \mathrm{d}x}$$

have you tried bring the (1+x^n)^(1/n) to the top? It would become (1+x^n)^n.

You've left the dx at the bottom
but how will it become $(1+x^n)^{n}$??? Bringing it to the top will change the sign of the exponent, right?

ideasrule
Homework Helper
You've left the dx at the bottom
but how will it become $(1+x^n)^{n}$??? Bringing it to the top will change the sign of the exponent, right?

Yes, I got confused. Sorry about that.

I tried letting xn = t, but that ended up with $$\frac{1}{n} \int \frac{1}{\sqrt[n]{t^2+t}}\:\mathrm{d}t$$, And I don't see how to do it.
Then I tried letting xn+1 = tn, and got something similarly unsolvable. Can anyone tell me what's the right substitution in this case?

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Char. Limit
Gold Member
Here's something:

$$\int x^{-n}(1+x^n)^{-1/n} \mathrm{d}x$$

And integrate by parts from there. Dunno if it works, though. I tried it on W-A and they used a weird substitution within a substitution.

I did that, letting (1+xn)-1/n as first function, and I ended up with :
$$\frac{x^{1-n}}{1-n}\,(1+x^n)^{-1/n} + \frac{x}{1-n} + \frac{x^{n+1}}{1-n^2} + C$$
Is that correct?

Char. Limit
Gold Member
Maybe. I'm too tired to check now. It looks right.

Will anyone please confirm if my answer is correct or not? This problem's been bugging me for quite some time.

try trigonometric substitution

Hello!
What trig substitution can you make? PM me if the OP wants to work it out themselves.
Thanks!