## Problem:

The method of power substitution assumes that you are familiar with the method of ordinary u-substitution and the use of differential notation. Integrate the following using the method of substitution.

\[\int \frac{1}{1 + {\sqrt{x}}} dx \]

## Solution:

\[x = u^2\]

\[\sqrt{x} = \sqrt{u^2} = u\]

\[dx = (2u)du\]

\[\int \frac{1}{1 + {\sqrt{x}}} dx = \int \frac{1}{1 + u} (2u)du\]

\[ \int \frac{2u}{1 + u} du\]

\[ \int (2 - \frac{2}{1 + u} )du\]

\[ \int (2 - 2\frac{1}{1 + u} )du\]

\[2u - 2ln|u + 1| + C\]

\[2\sqrt{x} - 2ln|\sqrt{x} + 1| + C\]