# Integration

• Apr 28th 2011, 10:48 PM
userod
Integration
Hi there can anyone help me integrate the following
$\int$ (x^2+1)/(x^4+1)

I've been trying unsuccessfully :(
• Apr 28th 2011, 10:54 PM
pickslides
What have you tried?

Maybe break it up using partial fractions?
• Apr 28th 2011, 11:13 PM
TheCoffeeMachine
$I = \int\frac{x^2+1}{x^4+1}\;{dx} = \int\frac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}}\;{dx } = \int\frac{1+\frac{1}{x^2}}{(x-\frac{1}{x})^2+2}\;{dx}$. Let t = x-1/x, then:

$I = \int\frac{1}{t^2+2}\;{dt}$ which hopefully is an integral that you are familiar with! :)
• Apr 28th 2011, 11:14 PM
userod
Ya that's what I've been trying I've $x^4+1$ factored as follows
(x^2+(2^1/2)x+1)(x^2-(2^1/2)x+1)
I'm having trouble solving the partial fractions though. Any ideas?
• Apr 28th 2011, 11:33 PM
Prove It
x^4 + 1 doesn't factorise... You should follow TheCoffeeMachine's method.
• Apr 28th 2011, 11:46 PM
userod
Great thanks ya I know that one inverse tan over root 2 all over root 2 right??!!! Thanks again guys really helpful site!
• Apr 28th 2011, 11:59 PM
TheCoffeeMachine
Quote:

Originally Posted by userod
Ya that's what I've been trying I've factored as follows
(x^2+(2^1/2)x+1)(x^2-(2^1/2)x+1)
I'm having trouble solving the partial fractions though. Any ideas?

Yes (I'm full of ideas this morning, haha), note that (x^2+√2x+1)+(x^2-√2x+1) = 2(x^2+1), so that:

\begin{aligned} I & = \int\frac{x^2+1}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}\;{dx} = \frac{1}{2}\int\frac{(x^2+\sqrt{2}x+1)+(x^2-\sqrt{2}x+1)}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}\;{dx} \\& = \frac{1}{2}\int\frac{1}{x^2-\sqrt{2}x+1}\;{dx}+\frac{1}{2}\int\frac{1}{x^2+ \sqrt{2}x+1}\;{dx} = I_{1}+I_{2}, ~ \text{say}. \end{aligned}

$I_{1} = \frac{1}{2}\int\frac{1}{\left(x-\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{\sqrt{2 }}\right)^2}\;{dx} = \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x-1\right)+k_{1}$ and

$I_{2} = \frac{1}{2}\int\frac{1}{\left(x+\frac{1}{\sqrt{2}} \right)^2+\left(\frac{1}{\sqrt{2}}\right)^2}\;{dx} = \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x+1\right)+k_{2}$, thus:

$I = \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x-1\right)+ \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x+1\right)+k.$
• Apr 29th 2011, 04:39 AM
mr fantastic
Quote:

Originally Posted by pickslides
What have you tried?

Maybe break it up using partial fractions?

Noting of course that $x^4 + 1 = (x^4 + 2x^2 + 1) - 2x^2 = (x^2 + 1)^2 - 2x^2 = (x^2 + 1 - \sqrt{2}x)(x^2 + 1 + \sqrt{2}x)$.