Evaluate the indefinite integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^x\sqrt[\phantom{2}]{2+e^x}dx$

Step $1$

We must decide what to choose for $u.$

If $u=f(x),$ then $du=f^{\text{'}}(x)dx,$ and so it is helpful to look for some expression in ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^x\sqrt[\phantom{2}]{2+e^x}dx$ for which the derivative is also present.

We see that $2+e^x$ is part of this integral, and the derivative of $2+e^x$ is $,$ which is also present.