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January 2018
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Problem:
The method of power substitution assumes that you are familiar with the method of ordinary usubstitution and the use of differential notation. Integrate the following using the method of substitution.
\[\int \frac{1}{1 + {\sqrt{x}}} dx \]
Solution:
\[x = u^2\]
\[\sqrt{x} = \sqrt{u^2} = u\]
\[dx = (2u)du\]
\[\int \frac{1}{1 + {\sqrt{x}}} dx = \int \frac{1}{1 + u} (2u)du\]
\[ \int \frac{2u}{1 + u} du\]
\[ \int (2  \frac{2}{1 + u} )du\]
\[ \int (2  2\frac{1}{1 + u} )du\]
\[2u  2lnu + 1 + C\]
\[2\sqrt{x}  2ln\sqrt{x} + 1 + C\]
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