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JH
Numerade Educator

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Problem 83 Medium Difficulty

The functions $ y = e^{x^2} $ and $ y = x^2 e^{x^2} $ don't have elementary antiderivatives, but $ y = (2x^2 + 1) e^{x^2} $ does. Evaluate $ \int (2x^2 + 1) e^{x^2}\ dx $.

Answer

$x e^{x^{2}}+C$

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Video Transcript

So here we are to evaluate this following in the girl. Let's just start off by rewriting it. So I'm just distributing the even X square. Now here, let me rewrite this as X times and then two ex either the X Square plus and then one times either that square. Now notice that if I called this f And if I called this G here then we have f prime equals one and we have g prime is two x e to the x square. So we cf prime here and then we see ji prime here. So we noticed that this is coming from the product rule. So the right hand side is what we have right now in this intolerant and so we'LL go ahead and write this in this form here someone in our f NRG. So that's her f and then RG is either the X Square and then we also have, of course, when he prime out there and then we use the fundamentals they remove Kokkalis, which allows us to cancel out the derivative with Ian a girl here as long as we go ahead and and that constant of integration c at the variant. And that's your final answer