Integrating Complexity: Solving \int \frac{1}{x^n(1+x^n)^{1/n}} \;\mathrm{d}x}

[Rats_N_Cats]

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.

 

[ideasrule]

Fixed your latex:

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

 

[Rats_N_Cats]

Fixed your latex:

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

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]

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.

 

[Rats_N_Cats]

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?

 

[Char. Limit]

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.

 

[Rats_N_Cats]

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]

Maybe. I'm too tired to check now. It looks right.

 

[Rats_N_Cats]

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

 

[nobahar]

Since you only want your answer checked, use Wolfram. It gives two substitutions in the method:

http://www.wolframalpha.com/input/?i=integrate+1/(x^(n)(1+x^(n))^(1/n)

 

[Zayer]

try trigonometric substitution

 

[nobahar]

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