Originally Posted by

**Furyan** Hello I'm sorry to ask again, but would somone be willing to look over this and tell me where I've gone wrong. Thank you.

I know the method I'm using is probably very shoddy, but it's the best I could do.

The question is integrate the following using the given substitution.

$\displaystyle \int\dfrac{\sqrt{x^2 + 4}}{x} dx$

$\displaystyle u^2 = x^2 + 4$

I got $\displaystyle \dfrac{dx}{du} = \dfrac{u}{x}$

I substituted these in first and got:

$\displaystyle \int(\dfrac{u}{x})(\dfrac{u}{x}) du$

$\displaystyle \int\dfrac{u^2}{x^2}$

Then using $\displaystyle u^2 = x^2 + 4$, I substituted again and got.

$\displaystyle \int\dfrac{u^2}{u^2 - 4}$

I didn't know how to integrate that so I used partial fractions and got:

$\displaystyle \int1 + \dfrac{1}{u - 2} - \dfrac{1}{u + 2}$

$\displaystyle u + \ln(u - 2) - \ln(u + 2) + c$

Using $\displaystyle u = \sqrt{x^2+4}$, I got:

$\displaystyle \sqrt{x^2 + 4} + \ln\dfrac{\sqrt{x^2+4} -2}{\sqrt{x^2 + 4} + 2} + c$

The given answer is:

$\displaystyle \sqrt{x^2 + 4} - \ln\dfrac{\sqrt{x^2+4} + 2}{\sqrt{x^2 - 4} - 2} + c$, which according to my graphing calculator is very different. I understand that this might be too much too ask, especially as I have only shown an outline of my working, but if anyone one can see an obvious place we're I've gone wrong I'd really appreciate it. My graphing calulator tells me the partial fractions are correct, so I really don't see what is wrong with what I have done.

Thank you.