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How do you integrate #(x^3)(e^(x^2))dx#?

Answer 1

Cameron Alexander

#1/2(x^2e^(x^2) - e^(x^2)) + C#

Use substitution method by considering #x^2 =u#, so that it is #x\ dx= 1/2\ du#.

The given integral is thus transformed to #1/2ue^u\ du#. Now integrate it by parts to have #1/2(ue^u-e^u) +C#.

Now substitute back #x^2# for u, to have the Integral as

#1/2(x^2e^(x^2) - e^(x^2)) + C#

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Answer 2

Aurora Alvarado

To integrate the expression (\int x^3 e^{x^2} , dx), you can use the substitution method. Let (u = x^2), then (du = 2x , dx). Substitute (u) and (du) into the integral to get (\frac{1}{2} \int u e^u , du). Now, integrate by parts with (dv = e^u , du) and (u = u). This yields (\frac{1}{2}(u e^u - \int e^u , du)). Integrating (e^u) with respect to (u) gives (\frac{1}{2}(u e^u - e^u) + C), where (C) is the constant of integration. Finally, substitute (u = x^2) back into the expression to get (\frac{1}{2}(x^2 e^{x^2} - e^{x^2}) + C).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.