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

1 Answer
Dec 17, 2016

Answer:

#(-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#