Question

\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]

Answer 1

\[\text{ Let I }= \int\frac{\left( 2x + 3 \right) dx}{\sqrt{x^2 + 4x + 5}}\]
\[ = \int\frac{\left( 2x + 4 - 1 \right)}{\sqrt{x^2 + 4x + 5}}dx\]
\[ = \int\frac{\left( 2x + 4 \right) dx}{\sqrt{x^2 + 4x + 5}} - \int\frac{dx}{\sqrt{x^2 + 4x + 5}}\]
\[ = \int\frac{\left( 2x + 4 \right) dx}{\sqrt{x^2 + 4x + 5}} - \int\frac{dx}{\sqrt{\left( x + 2 \right)^2 + 1}}\]
\[\text{ Consider, }\]
\[ x^2 + 4x + 5 = t\]
\[ \Rightarrow \left( 2x + 4 \right) dx = dt\]
\[ \therefore I = \int\frac{dt}{\sqrt{t}} - \int\frac{dx}{\sqrt{\left( x + 2 \right)^2 + 1^2}}\]
\[ = \int t^{- \frac{1}{2}} dt - \int\frac{dx}{\sqrt{\left( x + 2 \right)^2 + 1^2}}\]
\[ = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} - \text{ log }\left| x + 2 + \sqrt{\left( x + 2 \right)^2 + 1} \right| + C\]
\[ = 2\sqrt{x^2 + 4x + 5} - \text{ log }\left| x + 2 + \sqrt{x^2 + 4x + 5} \right| + C\]