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).
It helps you narrow down to a smaller answer for example there is pi, (3.14159265359) and there is more and more to that number. It is now estimated to 3.14 because it would be almost impossible to work with pi if it wasn't estimated
33.51 X 85% (0.85) = 28.48
