How would you evaluate the following the indefinite integral? (use C for the constant integration)

#int e^xsqrt((41 + e^x))dx#

1 Answer
May 12, 2018

#inte^xsqrt(41+e^x)dx=2/3(41+e^x)^(3/2)+C#

Explanation:

This can be solved using a #u-#substitution.

Let #u=41+e^x#

Calculating its differential, we get

#(du)/dx=e^x#
#du=e^xdx#

And this differential #du# does show up in our integral #intcolor(red)(e^x)sqrt(41+e^x)color(red)dx#, so the substitution is valid.

Rewriting with the substitution, we get

#intsqrtudu=intu^(1/2)du=2/3u^(3/2)+C#

Rewriting in terms of #x#, we get

#inte^xsqrt(41+e^x)dx=2/3(41+e^x)^(3/2)+C#