I am getting a sign error when evaluating:
$$ \int \dfrac {1} {\sqrt{-x^{2} – 4x}}dx$$
I completed the square in the denominator leaving me:
$$\int \dfrac {1} {\sqrt{-x^{2} – 4x + 4 – 4}}dx$$
$$\int \dfrac {1} {\sqrt{-(x^{2} + 4x – 4 + 4)}}dx$$
$$\int \dfrac {1} {\sqrt{-(x+2)^{2} +4}}dx$$
I then let $ u = x+2 , du = dx$, and $a = 2.$
$$\int \dfrac {du} {\sqrt{-u^{2} + a^{2}}}$$
$$\arcsin \dfrac {-(x+2)} {2} + C$$
However, the correct answer should be
$$\arcsin \dfrac {x+2} {2} + C$$
Where did I go astray?
Best Answer
$\displaystyle \int \dfrac{du}{\sqrt{a^2 - u^2}} = \arcsin \dfrac{u}{a} + C$