# How do you find the definite integral of #x * e^ (-x^2) dx# from #[1, sqrt2]#?

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To find the definite integral of (x \cdot e^{-x^2}) from (1) to (\sqrt{2}):

- Recognize that this integral requires substitution. Let (u = -x^2), hence (du = -2x dx).
- Adjust the integral bounds accordingly. When (x = 1), (u = -1), and when (x = \sqrt{2}), (u = -2).
- Rewrite the integral in terms of (u).
- Integrate the expression with respect to (u).
- Substitute back the original variable (x).
- Evaluate the expression at the upper and lower bounds and subtract.

The result is the definite integral of (x \cdot e^{-x^2}) from (1) to (\sqrt{2}).

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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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