Integrate $\frac{3}{\sqrt{-5x^2-4\sqrt{5}x+2}}$
For this question I first factored out the 3, seeing as its a constant
$$3\int\frac{1}{\sqrt{-5x^2-4\sqrt{5}x+2}}dx$$
I then noticed that this integral has a similar form to the integral of $\arcsin(x)$
$$\int\frac{1}{\sqrt{a^2-x^2}}dx=\arcsin(\frac{x}{a})+C, \vert x \vert \lt a$$
The similarities between these two integrals is clear, but I dont know how to get from $$-5x^2-4\sqrt{5}x+2$$ to $$a^2-x^2$$
Any help or ideas would be highly appreciated!
Best Answer
A rather messy completing the square:
$-5x^2-4\sqrt{5}x+2$
$-5\left(x^2+\frac{4\sqrt{5}x}{5}\right)+2$
$2-5\left(x+\frac{2\sqrt{5}}{5} \right)^2+4$
$(\sqrt{6})^2-\left(\sqrt{5}\left(x+\frac{2\sqrt{5}}{5} \right)\right)^2$