How Do You Integrate 6x^2e^(x^3) in Differential Equations?

[montana111]

Homework Statement

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 I am doing something wrong or there is a trick I am missing but I've 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.

 

[HallsofIvy]

Did you notice that you have
x^2e^{x^2}
in your title but the integral you want is of
x^2e^{x^3}?

I mention that because I suspect that x^2e^{x^3} can't be integrated in any simple form while x^2e^{x^3} is easy!

Let u= x^3 and then dy= 3x^2dx so your integral becomes
\int x^2e^{x^3}dx= \frac{1}{3}\int e^{x^3}\left(3x^2dx\right)= \frac{1}{3}\int e^u du

 

[montana111]

wow. I am dumb. thank you.

p.s. where on the site can i see how to make tags for actual mathmatical symbols like you have in your reply?

 

[Дьявол]

wow. I am dumb. thank you.

p.s. where on the site can i see how to make tags for actual mathmatical symbols like you have in your reply?

When you open the Advanced Options of the post, you got the \Sigma button where you can choose the mathematics symbol. This forum has implemented LaTeX, so you can write the formulas inside the tag [tеx][\tex]