# How do you integrate #int e^xsqrt(1-e^x)dx#?

Given the existence of a square root, any addition to the root will necessitate the application of a chain rule, making this problem challenging. Using the substitution, we will make:

Consequently, following the substitution, the final integral is

After applying the integral, multiply the resultant number by the exponent, and divide it.

Once you replace the u from the initial scenario, you have your definitive response.

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To integrate ( \int e^{x\sqrt{1-e^x}} , dx ), you can use the substitution method. Let ( u = \sqrt{1 - e^x} ), then ( u^2 = 1 - e^x ) and ( e^x = 1 - u^2 ). Taking the derivative with respect to ( x ) gives ( du/dx = -\frac{1}{2\sqrt{1 - e^x}}e^x ), which simplifies to ( du = -\frac{1}{2u}e^x , dx ).

Substituting these expressions into the integral, we get: [ \int e^{x\sqrt{1-e^x}} , dx = -2\int e^{-x} , du ]

This is now a simpler integral to evaluate. Integrating ( e^{-x} ) with respect to ( u ) gives ( -e^{-x} ), so: [ -2\int e^{-x} , du = -2(-e^{-x}) + C = 2e^{-x} + C ]

Thus, the integral of ( \int e^{x\sqrt{1-e^x}} , dx ) equals ( 2e^{-x} + C ), where ( C ) is the constant of integration.

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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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