Question

integrate dx/√x (1+√x)

Answer 1

\[\int\limits_{}^{} \frac {dx}{ \sqrt {x} ( 1 + \sqrt {x})}\]\[u = 1 + \sqrt {x}, du = \frac {dx}{2 \sqrt {x}}, 2du = \frac {dx}{\sqrt {x}}\]\[\int\limits\limits_{}^{} \frac {dx}{ \sqrt {x} ( 1 + \sqrt {x})} \rightarrow \int\limits_{}^{} \frac {2du}{u}\]\[2 \int\limits_{}^{} \frac {du}{u} = 2 \ln \left| u \right| + C\]Substitute u back, and we get \[\int\limits\limits_{}^{} \frac {dx}{ \sqrt {x} ( 1 + \sqrt {x})} = 2 \ln \left| 1 + \sqrt {x} \right| + C\]