# How do you evaluate the integral #int xe^(-x^2)#?

The answer is

We perform this integral by substitution

Therefore,

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To evaluate the integral (\int xe^{-x^2} , dx), we use a substitution method. Let (u = -x^2), then (du = -2x , dx). Solving for (x , dx) gives us (x , dx = -\frac{1}{2} du). Substituting these into the integral, we get (-\frac{1}{2} \int e^u , du). Integrating (e^u) with respect to (u) gives us (-\frac{1}{2} e^u + C), where (C) is the constant of integration. Substituting (u = -x^2) back in, we get (-\frac{1}{2} e^{-x^2} + C). So, (\int xe^{-x^2} , dx = -\frac{1}{2} e^{-x^2} + C).

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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.

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.

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.

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