Question

\[\int\frac{1}{4 x^2 + 4x + 5} dx\]

Answer 1

\[\text{ Let I }= \int\frac{dx}{4 x^2 + 4x + 1 + 4}\]
\[ = \int\frac{dx}{\left( 2x \right)^2 + 2 \times 2x + 1 + 22}\]
\[ = \int\frac{dx}{\left( 2x + 1 \right)^2 + 2^2}\]
\[\text{ Putting }\left( 2x + 1 \right) = t\]
\[ \Rightarrow 2 \text{ dx = dt }\]
\[ \Rightarrow dx = \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\frac{dt}{t^2 + 2^2}\]
\[ = \frac{1}{2} \times \frac{1}{2} \text{ tan}^{- 1} \left( \frac{t}{2} \right) + C\]
\[ = \frac{1}{4} \text{ tan}^{- 1} \left( \frac{2x + 1}{2} \right) + C ....................\left[ \because t = \left( 2x + 1 \right) \right]\]