How do you integrate #int e^xsqrt(1-e^x)dx#?
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
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:
Therefore, the final intergral after the substitution is
Then when you apply the intergral, you add the exponent and then divide the number that you get,
Substitute back the u from the original situation and you have your final answer