Find an antiderivative?

Question

Find an antiderivative of f(x)=e^(x+e^x).

 

Answer: e^e^x.

 

Here's the work:

f(x)=e^x*e^e^x

u=e^x

du=e^x dx

 

How should I set up the integral with this u-substitution? Please show all your work.

Answer 1

Use mental substitution,

∫e^(x+e^x)dx

= ∫e^x * e^(e^x) dx

= ∫e^(e^x) de^x

= e^(e^x) <==Answer

Answer 2

∫ e^u du = e^u +c

 

 

Here U = X +e ^x

 

        du = (1 + e^x ) dx

 

         ∫ u du = ∫ e^{( x +e^x )} ( 1 + e^x) dx =e ^{(x + e^x)} +C