# Thread: Stuck on a tricky integral

1. ## Stuck on a tricky integral

I tried get a tidy image of the integral with Integrator, but it told me the answer does not exist. I'm almost certain it does however.

(1+ln(x))*sqrt(1+(xln(x))^2)

Help! I tried integration by parts but it turns into a jumbled mess, unless I'm overlooking something.

2. Wait a second, I just had some inspiration. I'll let you know If I truley need help...

3. Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?

4. Originally Posted by phack
Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?
Good job spotting the substitution. Now what you want to do is let $\displaystyle u = sinh(y)$. Then $\displaystyle du = cosh(y)dy$, so your integral turns into:
$\displaystyle \int (1 + xln(x)) \sqrt{1 + (x ln(x))^2)} dx = \int \sqrt{1 + u^2} \, du =$$\displaystyle \int \sqrt{1 + sinh^2(y)} \cdot cosh(y)dy = \int cosh^2(y) dy$

$\displaystyle = \int \frac{1}{4}(e^{2y} + e^{-2y} + 2)dy = ...$

You can finish this from here.

-Dan