How to Integrate 1/x+x sqrt(x)

 

\begin{array}{l} \smallint \frac{{\bf{1}}}{{{\bf{x}} + {\bf{x}}\sqrt {\bf{x}} }}{\rm{d}}{\bf{x}}\\\\ By\;Substitution\\ \;\;\;\;\;\;\;\;\;\;Suppose \Rightarrow {z^2} = x\;\\ 2zdz = dx\\ \smallint \frac{{2\;{\rm{z}}}}{{{z^2} + {z^3}}}{\rm{d}}x\\ \smallint \frac{{2\;}}{{z({z^\;} + 1)}}{\rm{d}}x\\ By\;Using\;Partial\;Fraction\\ \smallint \frac{2}{{z({z^\;} + 1)}}{\rm{d}}x = \frac{A}{z} + \frac{B}{{z + 1}}\;\;\\ \;\;\;\;\;\;\;\;\;Let \Rightarrow z = 0 \Rightarrow A = 2\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Let \Rightarrow z = - 1 \Rightarrow B = - 2\\ \smallint \frac{2}{z} - \frac{2}{{z + 1}}{\rm{d}}x\\ 2\ln \left| z \right| - 2\ln \left| {z + 1} \right| + c\\ \Rightarrow 2\ln \left| {\sqrt x } \right| - 2\ln \left| {\sqrt x + 1} \right| + c \end{array}