<s>[INST] Definite integral for a 4 degree function 
The integral is:
  $$\int_0^a \frac{x^4}{(x^2+a^2)^4}dx$$

I used an approach that involved substitution of x by $a\tan\theta$. No luck :\ . Help?
 [/INST] One way to solve this problem is to do integration by parts.
$$\int{x^3 \frac{x}{(x^2+a^2)^4}dx}=-\frac{x^3}{6(x^2+a^2)^3}+\frac{1}{2}\int{x\frac{x}{(x^2+a^2)^3} dx}.$$
Continuing further
$$\int{x\frac{x}{(x^2+a^2)^3} dx} = -\frac{1}{4} \frac{x}{(x^2+a^2)^2}+\frac{1}{4}\int{\frac{1}{(x^2+a^2)^2}dx}.$$
Now you can substitute $x=a\tan{\theta}$ to get the final result.
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