Question

Hey guys can you give multiple ways to solve this ?!
integrate With respext to x :
1/(1+(x^4))

Answer 1

\[\int\limits_{}^{}\frac{ 1 }{ 1+ x^4 } dx\]

Answer 2

Residues would be the best and simplest way for this..

Answer 3

... if it were a definite integral, sure.

Try partial fractions:
\[x^4+1=-(x^2-\sqrt2x+1)(x^2+\sqrt2x+1)\]
Then
\[\begin{align*}-\frac{1}{(x^2-\sqrt2x+1)(x^2+\sqrt2x+1)}&=\frac{Ax+B}{x^2-\sqrt2x+1}+\frac{Cx+D}{x^2+\sqrt2x+1}\\\\
-1&=Ax^3+Bx^2+\sqrt2Ax^2+\sqrt2Bx+Ax+B\\
&~~~~~~~~~~~+Cx^3+Dx^2-\sqrt2Cx^2-\sqrt2Dx+Cx+D\\\\
-1&=(A+C)x^3+(\sqrt2A+B-\sqrt2C+D)x^2\\
&~~~~~~~~~~~+(A+\sqrt2B+C-\sqrt2D)x+B+D
\end{align*}\]
So you have
\[\begin{cases}\begin{align*}A+C&=0\\\sqrt2A+B-\sqrt2C+D&=0\\A+\sqrt2B+C-\sqrt2D&=0\\
B+D&=-1\end{align*}\end{cases}\]
Solutions for the constants (5th column of matrix, top to bottom gives A,B,C,D):
http://www.wolframalpha.com/input/?i=rref+%7B%7B1%2C0%2C1%2C0%2C0%7D%2C%7BSqrt%5B2%5D%2C1%2C-Sqrt%5B2%5D%2C1%2C0%7D%2C%7B1%2CSqrt%5B2%5D%2C1%2C-Sqrt%5B2%5D%2C0%7D%2C%7B0%2C1%2C0%2C1%2C-1%7D%7D

Answer 4

So now you're integrating
\[\frac{1}{2\sqrt2}\int\frac{x-\sqrt2}{x^2-\sqrt2x+1}~dx-\frac{1}{2\sqrt2}\int\frac{x+\sqrt2}{x^2+\sqrt2x+1}~dx\]

Answer 5

thanks guys for your helpful repiles ! If someone gets another way ... just type it !! You guys stay awesome!!