Integral of [(6x^2)(e^(x^2))]

  • Thread starter montana111
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  • #1
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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 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.
 

Answers and Replies

  • #2
HallsofIvy
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Did you notice that you have
[tex]x^2e^{x^2}[/tex]
in your title but the integral you want is of
[tex]x^2e^{x^3}[/tex]?

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

Let [itex]u= x^3[/itex] and then [itex]dy= 3x^2dx[/itex] so your integral becomes
[tex]\int x^2e^{x^3}dx= \frac{1}{3}\int e^{x^3}\left(3x^2dx\right)= \frac{1}{3}\int e^u du[/tex]
 
  • #3
12
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wow. im 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?
 
  • #4
365
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wow. im 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 [tex]\Sigma[/tex] button where you can choose the mathematics symbol. This forum has implemented LaTeX, so you can write the formulas inside the tag [tеx][\tex]
 

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