# Thread: Indefinite Integration

1. ## Indefinite Integration

I've got another integral that I can't solve. =/ I know it looks really ugly but I'm certain it's correct.

$\large \int \frac{kx^{2}+u}{ln\left |x \right |-\sqrt{\frac{1}{ku}x^{4}+k}}dx$

If I've missed something really obvious I'd appreciate it if you gave me a hint and not the full solution. Thanks in advance.

2. ## Re: Indefinite Integration

No, there is nothing "really obvious". That is, as you say, an ugly integral and it cannot be done in any simple way.

3. ## Re: Indefinite Integration

Trigonometric substitution seems out the window because of the ln|x| in the denominator.

Doing it by parts doesn't seem to work.

$\large \int \frac{kx^{2}+u}{ln\left |x \right |-\sqrt{\frac{1}{ku}x^{4}+k}}dx$
$\large \int kx^{2}+udx\left ( \frac{1}{ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k}} \right ) -\int \left [\int \left (kx^{2}+u \right )dxDx\left ( \frac{1}{ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k}} \right ) \right ]dx$

Now looking only at this part of the second term...

$\large Dx\left ( \frac{1}{ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k}} \right )$
$\large =Dx\left [\left ( ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k} \right )^{-1} \right ]$
$\large =-1\left ( ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k} \right )^{-2}\times Dx\left ( ln\left | x \right |-\sqrt{\frac{1}{ku}x^{4}+k} \right ) \right )$

etc. So the second integral doesn't look like it can be simplified into anything workable.

So the solution must concern normal u-substitution or some method of integration that I'm not familiar with. I was looking for another method of integration and I read about something called 'Integration by Using Parametric Differentiation', I didn't really understand the article so perhaps that's necessary here?

4. ## Re: Indefinite Integration

I assume we are integrating with respect to x - What are k and u? Not the actual values, but are they supposed to be differentiable functions of a variable, or constants, or . . . You need this information to integrate

5. ## Re: Indefinite Integration

k and u are both constants, and yes - the aim is to integrate in terms of x.