# Math Help - Evaluate definite integral

1. ## Evaluate definite integral

Hi!

Problem: $\int_{1}^{2} \frac{x\cdot ln(x)}{(1+x^{2})^{2}} \; dx$

What is the smartest approach?
I tried letting $(1+x^{2})=t$ , because then I found that $x = \sqrt{t-1} \mbox{ and } dx=\frac{dt}{2\sqrt{t-1}}$ , so that square root expression would cancel which is nice.

I got stuck somewhere along the road though.

I landed first at: $\frac{1}{4} \int_{2}^{5} \frac{ln(t-1)}{t^{2}} \; dt$ and from here I tried using integration by parts first, integrating $\frac{1}{t^{2}}$ , but this might be wrong approach.

Thx!

2. $\int_{1}^{2}\frac{xln(x)}{(1+x^{2})^{2}}dx$

Using parts, we can let $u=ln(x), \;\ du=\frac{1}{x}dx, \;\ v=\frac{-1}{2(x^{2}+1)}, \;\ dv=\frac{x}{(x^{2}+1)^{2}}dx$

Putting it altogether we get:

$\frac{1}{2}\cdot\frac{ln(x)}{x^{2}+1}+\int\frac{1} {2x(x^{2}+1)}dx$