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]