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

  1. Jul 9, 2009 #1
    1. The problem statement, all variables and given/known data
    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.
     
  2. jcsd
  3. Jul 9, 2009 #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]
     
  4. Jul 9, 2009 #3
    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?
     
  5. Jul 10, 2009 #4
    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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