Integrate: x*3^x^2dx

I am assuming you want to integrate:
∫ x*3^(x^2) dx.

Start out by substituting:
u = x^2 ==> du = 2x dx.

Then, we have:
∫ x*3^(x^2) dx
= 1/2 ∫ 3^(x^2) (2x dx), by re-writing
= 1/2 ∫ 3^u du, by applying substitutions
= 3^u/[2ln(3)] + C, by integrating
= 3^(x^2)/[2ln(3)] + C, by back-substituting u = x^2.

I hope this helps!