\(\int \frac{1}{4x^2 +4x+5}dx\)
Answer & explanation
Correct answer: option 2
\(\begin{aligned}4x^2+4x+5&=4\left[\left(x+\frac{1}{2}\right)^{2}+1\right]\\ \int \frac{1}{4x^2+4x+5}dx&=\frac{1}{4}\int \frac{dx}{\left(x+\frac{1}{2}\right)^{2}+1}\\ &=\frac{1}{4}\tan^{-1}\left(x+\frac{1}{2}\right)+c\end{aligned}\)