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

1 Answer
Dec 17, 2016

(-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