Weird Integral

  • #1

Homework Statement



An Integral : [tex]

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

[/tex]

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 :frown: so I can't even start it. If anyone shows me the first step I'll try to take it from there.
 
Last edited:

Answers and Replies

  • #2
ideasrule
Homework Helper
2,271
0
Fixed your latex:

[tex]
\int \frac{1}{x^n(1+x^n)^{1/n}} \mathrm{d}x
[/tex]
 
Last edited:
  • #3
Fixed your latex:

[tex]
\int \frac{1}{x^n(1+x^n)^{1/n} \mathrm{d}x}
[/tex]

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 :wink:
but how will it become [itex](1+x^n)^{n}[/itex]??? Bringing it to the top will change the sign of the exponent, right?
 
  • #4
ideasrule
Homework Helper
2,271
0
You've left the dx at the bottom :wink:
but how will it become [itex](1+x^n)^{n}[/itex]??? Bringing it to the top will change the sign of the exponent, right?

Yes, I got confused. Sorry about that.
 
  • #5
I tried letting xn = t, but that ended up with [tex]
\frac{1}{n} \int \frac{1}{\sqrt[n]{t^2+t}}\:\mathrm{d}t
[/tex], 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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  • #6
Char. Limit
Gold Member
1,208
14
Here's something:

[tex]\int x^{-n}(1+x^n)^{-1/n} \mathrm{d}x[/tex]

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.
 
  • #7
I did that, letting (1+xn)-1/n as first function, and I ended up with :
[tex]
\frac{x^{1-n}}{1-n}\,(1+x^n)^{-1/n} + \frac{x}{1-n} + \frac{x^{n+1}}{1-n^2} + C
[/tex]
Is that correct?
 
  • #8
Char. Limit
Gold Member
1,208
14
Maybe. I'm too tired to check now. It looks right.
 
  • #9
Will anyone please confirm if my answer is correct or not? This problem's been bugging me for quite some time.
 
  • #11
21
0
try trigonometric substitution
 
  • #12
495
2
Hello!
What trig substitution can you make? PM me if the OP wants to work it out themselves.
Thanks!
 

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