# Stuck on a tricky integral

• Feb 5th 2007, 01:21 PM
phack
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.
• Feb 5th 2007, 01:24 PM
phack
Wait a second, I just had some inspiration. I'll let you know If I truley need help...
• Feb 5th 2007, 01:45 PM
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 :confused: How do I integrate sqrt(1+u^2) ?
• Feb 5th 2007, 04:24 PM
topsquark
Quote:

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 :confused: 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