Integration

**userod** · April 28th 2011, 09:48 PM

Hi there can anyone help me integrate the following
$\int$ (x^2+1)/(x^4+1)

I've been trying unsuccessfully

**pickslides** · April 28th 2011, 09:54 PM

What have you tried?

Maybe break it up using partial fractions?

**TheCoffeeMachine** · April 28th 2011, 10:13 PM

$I = \int\frac{x^2+1}{x^4+1}\;{dx} = \int\frac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}}\;{dx } = \int\frac{1+\frac{1}{x^2}}{(x-\frac{1}{x})^2+2}\;{dx}$ . Let t = x-1/x, then:

$I = \int\frac{1}{t^2+2}\;{dt}$ which hopefully is an integral that you are familiar with! :)

**userod** · April 28th 2011, 10:14 PM

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?

**Prove It** · April 28th 2011, 10:33 PM

x^4 + 1 doesn't factorise... You should follow TheCoffeeMachine's method.

**userod** · April 28th 2011, 10:46 PM

Great thanks ya I know that one inverse tan over root 2 all over root 2 right??!!! Thanks again guys really helpful site!

**TheCoffeeMachine** · April 28th 2011, 10:59 PM

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:

$I = \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x-1\right)+ \frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}x+1\right)+k.$

**mr fantastic** · April 29th 2011, 03:39 AM

Originally Posted by pickslides

What have you tried?

Maybe break it up using partial fractions?

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

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