This is an example problem from my stewart calculus book.
the whole problem is "solve this diff eq: y' + 3(x^2)y = 6x^2
next step is to find I(x) which is e^(x^3) and multiply everything by that
then you say (I(x)y)' = 6(x^2)(e^(x^3)) and then you integrate both sides
so I(x)y = the integral of [6(x^2)(e^(x^3))].
this is where i have problems. i cannot figure out the integral on the right hand side.
the book then shows the answer to the integral as 2e^(x^3) + C
and the final answer is then y = Ce^(-x^3)
Also if someone could explain to me why the final answer has the negative sign in the exponent that would be helpful.
Ive looked at the problem for a while so maybe im doing something wrong or there is a trick im missing but ive tried to do it "by parts" with no luck. I was under the impression that you cannot just take the integral of an exponential like e^x^x.