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Math Help - Evaluate definite integral

  1. #1
    Senior Member Twig's Avatar
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    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!
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    Thanks
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    \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
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