Question

Integral of 3a^2/1+a^6

Answer 1

The trick here is to use an appropriate substitution.  Let u=a^3.
Then du/da=3a^2, and du=3a^2da.

We can now make two key substitutions:  In (3a^2)da/(1+a^6), replace 3a^2 by du and a^6 by u^2.

Then we have the integral of du/(1+u^2).

Integrating, we get arctan u + c.  Substituting a^3 for u, the final result (the integral in question) is    arctan a^3 + c.

Check this by differentiation.  if you find the derivative with respect to a of arctan a^3 + c, you MUST obtain the result 3a^2/(1+a^6).