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

Answer 1

Emily Ammerman

#1/2*(e^(-1)-e^(-2))#

#int_1^sqrt2 xe^(-x^2)*dx#

=#-1/2int_1^sqrt2 -2xe^(-x^2)*dx#

=#[-1/2e^(-x^2)]_1^sqrt2#

=#1/2*(e^(-1)-e^(-2))#

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Answer 2

Christopher Agresta

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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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.