Question

Evaluate:

$\int \frac{dx}{\sqrt{1+4x^2}}$.

Answer 1

Taking $\frac{1}{2}$ common factor

$$\begin{array}{c}\frac{1}{2}\int \frac{dx}{\sqrt{\left(\frac{1}{4}+x^2\right)}}\left[\int \frac{dx}{\sqrt{a^2+x^2}}=\log \left|x+\sqrt{a^2+x^2}\right|\right]\\ \frac{1}{2}\int \frac{dx}{\sqrt{{\left(\frac{1}{2}\right)}^2+x^2}}\\ \frac{1}{2}\log \left|x+\sqrt{\frac{1}{4}+x^2}\right|+C\\ \frac{1}{2}\log \left|x+\sqrt{\frac{1+4x^2}{2}}\right|+C\end{array}$$