# How do you find the definite integral of #2/(sqrt(x) *e^-sqrt(x))# from #[1, 4]#?

Simplify the negative exponent by bringing it to the numerator.

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To find the definite integral of ( \frac{2}{\sqrt{x} \times e^{-\sqrt{x}}} ) from 1 to 4, you can use substitution. Let ( u = \sqrt{x} ), then ( du = \frac{1}{2\sqrt{x}} dx ). The integral becomes ( 4\int_{1}^{2} e^u du ). Integrating ( e^u ) with respect to ( u ) yields ( e^u ). Evaluate this from 1 to 2, which gives ( 4(e^2 - e) ) or approximately ( 35.692 ).

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To find the definite integral of ( \frac{2}{\sqrt{x} \times e^{-\sqrt{x}}} ) from ( x = 1 ) to ( x = 4 ), you can use the substitution method.

Let ( u = \sqrt{x} ). Then ( du = \frac{1}{2\sqrt{x}} dx ), which can be rewritten as ( 2\sqrt{x} , du = dx ).

Now, substitute ( u ) and ( dx ) in terms of ( u ) into the integral:

( \int_{1}^{4} \frac{2}{\sqrt{x} \times e^{-\sqrt{x}}} , dx = \int_{u=1}^{u=2} \frac{2}{u \times e^{-u}} \times 2\sqrt{x} , du )

This simplifies to:

( \int_{1}^{2} \frac{4}{u \times e^{-u}} , du )

Now, this integral can be solved as a standard integral. After integration, evaluate the result from ( u = 1 ) to ( u = 2 ) to find the definite integral.

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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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