Question

Evaluate:
$\int \frac{1}{\sqrt{1-x-x^2}dx}$

Answer 1

$\begin{array}{c}Given,\\ \int \frac{1}{\sqrt{1-x-x^2}dx}\\ Now,\\ =\int \frac{dx}{\sqrt{\left(1-\frac{1}{4}-\left(x^2+x+\frac{1}{4}\right)\right)}}\\ =\int \frac{dx}{\sqrt{{\left(\frac{\sqrt{5}}{2}\right)}^2-{\left(x+\frac{1}{2}\right)}^2}}\\ ={Sin}^{-1}\left(\frac{2x+1}{\sqrt{5}}\right)+C\end{array}$