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

Dec 17, 2016

$\frac{- 2 {\left(1 - {x}^{e}\right)}^{\frac{3}{2}}}{3} + C$

#### Explanation:

There is one thing to remember when you first look at an integral problem involving substitution:
Which section looks like the derivative of the other.
Since we are dealing with exponental function, this is the simplest problem there is. The derivative of ${e}^{x} = {e}^{x}$.

Seeing that there is a square root, anything that is adding in the root will cause the equation to require a chain rule making this problem difficult therefore. With the substitution, we will make:

$u = 1 - {e}^{x}$
$\mathrm{du} = - {e}^{x} \mathrm{dx}$

Therefore, the final intergral after the substitution is

$- \int \sqrt{u} \mathrm{du}$
$- \int {u}^{\frac{1}{2}} \mathrm{du}$

Then when you apply the intergral, you add the exponent and then divide the number that you get,

$\frac{- 2 {u}^{\frac{3}{2}}}{3} + C$

Substitute back the u from the original situation and you have your final answer

$\frac{- 2 {\left(1 - {x}^{e}\right)}^{\frac{3}{2}}}{3} + C$