Question

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

Answer 1

`(-2(1-x^e)^(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`
`du=-e^x dx`

Therefore, the final intergral after the substitution is

`-intsqrt(u) du`
`-intu^(1/2)du`

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

`(-2u^(3/2))/3+C`

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

`(-2(1-x^e)^(3/2))/3+C`