So I have been having trouble with finding the indefinite integral of
1/ (1+x^4)^(1/4), by this I mean (1+x^4)^(-1/4).
I was given a hint that says to use the substitution t^4= 1+x^(-1/4).
Any help would be appreciated, and thanks in advance!
So I have been having trouble with finding the indefinite integral of
1/ (1+x^4)^(1/4), by this I mean (1+x^4)^(-1/4).
I was given a hint that says to use the substitution t^4= 1+x^(-1/4).
Any help would be appreciated, and thanks in advance!
Wolfram says that the solution involves the Hypergeometric Function.
http://www.wolframalpha.com/input/?i=integral[%281+%2B+x^4%29^%28-1%2F4%29]
I tell you
There exists a solution , perhaps the only one , to this integral
$\displaystyle \int \frac{dx}{ (1 + x^n)^{\frac{1}{n}}}$
Which is substituting $\displaystyle 1 + x^{-n} = t^n $ <- be cafeful
the power of $\displaystyle x $ is this time $\displaystyle -n$ not $\displaystyle n $
so
$\displaystyle -n x^{-n-1}dx = n t^{n-1} dt $
$\displaystyle dx = - x^{n+1} t^{n-1} dt $
$\displaystyle \int \frac{dx}{ (1 + x^n)^{\frac{1}{n}}}$
$\displaystyle \int \frac{1}{ (1 + x^n)^{\frac{1}{n}}} \cdot ( - x^{n+1} t^{n-1} dt ) $
$\displaystyle = - \int \frac{x^{n+1} t^{n-1}}{x (1 + x^{-n})^{\frac{1}{n}}} ~dt $
$\displaystyle - \int \frac{x^{n} t^{n-1}}{ t} ~dt $
$\displaystyle - \int \frac{t^{n-2}}{t^n - 1} ~dt $
When $\displaystyle n = 4 $
the integral is tansformed to
$\displaystyle - \int \frac{ t^2 }{ t^4 - 1}~dt $
$\displaystyle = -\frac{1}{2} \int \left [ \frac{1}{t^2 - 1} + \frac{1}{t^2 + 1} \right ]~dt $
You don't always have to believe to Wolfram.
In the integral put $\displaystyle x=\frac1z$ and it becomes $\displaystyle -\int\frac{dz}{z\sqrt[4]{1+z^4}},$ now put $\displaystyle t^4=1+z^4$ and the integral is $\displaystyle -\int\frac{t^2}{t^4-1}\,dt$ which is the same as simplependulum gave.