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I need to know the primitive function (Antiderivative) of this function: f(x)= $xe^{x^2+2x}$ without using integral please. Also, please how could I find the primitive functions of those kind like v(x)u(x) ? is there any technique concerning those types? Thanks in advance

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  • $\begingroup$ Rewrite as $(x+1)e^{x^2+2x}-e^{x^2+2x}$. The first has an easy antiderivative. The antiderivative of the second cannot be written as a closed form expression using elementary functions. $\endgroup$ Jan 29, 2016 at 18:39
  • $\begingroup$ @Dr.SonnhardGraubner What error? let me explain to you, the function in the exercise was $\left(x+1\right)e^{x^2+2x}$ so it's equals to : $xe^{x^2+2x}+e^{x^2+2x}$ the second one, I mean $e^{x^2+2x}$ is easy to find and it's primitive is : $\frac{1}{2x+2}e^{x^2+2x}$ but the other, I failed to find it. $\endgroup$ Jan 29, 2016 at 18:45
  • $\begingroup$ @AndréNicolas I didn't get it, why should I rewrite it like that? $\endgroup$ Jan 29, 2016 at 18:47
  • $\begingroup$ @AmineMarzouki It was written that way to exploit the fact that one of the primitives of $u'(x)e^{u(x)}$ is $e^{u(x)}$. $\endgroup$
    – Workaholic
    Jan 29, 2016 at 18:47
  • $\begingroup$ @Dr.SonnhardGraubner Ooops I did a mistake, it's wrong, sorry $\endgroup$ Jan 29, 2016 at 18:48

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From your comments, it seems that the original requested primitive was that of the function $f\colon x\mapsto(x+1)e^{x^2+2x}$. Decomposing it into $x e^{x^2+2x}$ and $e^{x^2+2x}$ wont yield a satisfying result, since the former will involve the error function. Instead, exploit the fact that the primitive of $u'(x)e^{u(x)}$ is $e^{u(x)}$. Here we can clearly see that $f(x)$ can be written as $\tfrac12(2x+2)e^{x^2+2x}$ which is $\tfrac{1}{2}(x^2+2x)'e^{x^2+2x}$. Can you now conclude what the primitive of $f$ is?

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