Question

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

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

Answer 1

`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`