More Examples of Integration

Example:

\int {\frac{{{{\text{x}}^2} + 2\sqrt {{\text{x}} - 1} }}{{2{{\text{x}}^2}\sqrt {{\text{x}} - 1} }}{\text{dx}}}

Solution:
We have

\int {\frac{{{{\text{x}}^2} + 2\sqrt {{\text{x}} - 1} }}{{2{{\text{x}}^2}\sqrt {{\text{x}} - 1} }}{\text{dx}}}


\begin{gathered} = \int {\left[ {\frac{{{{\text{x}}^2}}}{{2{{\text{x}}^2}\sqrt {{\text{x}} - 1} }} + \frac{{2\sqrt {{\text{x}} - 1} }}{{2{{\text{x}}^2}\sqrt {{\text{x}} - 1} }}} \right]{\text{dx}}} \\ = \int {\left[ {\frac{1}{{2\sqrt {{\text{x}} - 1} }} + \frac{{2\sqrt {{\text{x}} - 1} }}{{2{{\text{x}}^2}\sqrt {{\text{x}} - 1} }}} \right]{\text{dx}}} \\ = \int {\left[ {\frac{1}{{2\sqrt {{\text{x}} - 1} }} + \frac{1}{{{{\text{x}}^2}}}} \right]{\text{dx}}} \\ = \frac{1}{2}\int {{{\left( {{\text{x}} - 1} \right)}^{ - \frac{1}{2}}}{\text{dx}}} + \int {{{\text{x}}^{ - 2}}{\text{dx}}} \\ = \frac{1}{2}\frac{{{{\left( {{\text{x}} - 1} \right)}^{ - \frac{1}{2} + 1}}}}{{ - \frac{1}{2} + 1}} + \frac{{{{\text{x}}^{ - 2 + 1}}}}{{ - 2 + 1}} + {\text{c}} \\ = \frac{1}{2}\frac{{{{\left( {{\text{x}} - 1} \right)}^{\frac{1}{2}}}}}{{\frac{1}{2}}} - {{\text{x}}^{ - 1}} + {\text{c}} \\ = \sqrt {{\text{x}} - 1} - \frac{{\text{1}}}{{\text{x}}} + {\text{c}} \\ \end{gathered}