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