Question

\[\int\frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]

Answer 1

\[\text{ We  have,} \]
\[I = \int \frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx }\]
\[\text{ Putting  x }+ 2 = t^2 \]
\[ \Rightarrow x = t^2 - 2\]
\[\text{ Diff both sides }
\]
\[dx = 2t \text{ dt }\]
\[I = \int \frac{\left( t^2 - 2 + 1 \right)2t \text{ dt }}{\left( t^2 - 2 - 1 \right)t}\]
\[ = 2\int \left( \frac{t^2 - 1}{t^2 - 3} \right)dt\]
\[ = 2\int\left( \frac{t^2 - 3 + 2}{t^2 - 3} \right)dt\]
\[ = 2\int \left( \frac{t^2 - 3}{t^2 - 3} \right)dt + 4\int\frac{dt}{t^2 - 3}\]
\[ = 2\int dt + 4\int\frac{dt}{t^2 - \left( \sqrt{3} \right)^2}\]
\[ = 2t + 4 \times \frac{1}{2\sqrt{3}}\text{ log } \left| \frac{t - \sqrt{3}}{t + \sqrt{3}} \right| + C\]
\[ = 2\sqrt{x + 2} + \frac{2}{\sqrt{3}}\text{ log }\left| \frac{\sqrt{x + 2} - \sqrt{3}}{\sqrt{x + 2} + \sqrt{3}} \right| + C\]