Question

Evaluate $\int \frac{1}{\sqrt{7-6x-x^2}}$

Answer 1

$\int \frac{1}{\sqrt{7-6x-x^2}}dx$

$\int \frac{1}{\sqrt{-x^2-6x+7}}$

$\int \frac{dx}{\sqrt{-\left(x^2+6x-7\right)}}$

$\int \frac{dx}{\sqrt{-1\left[x^2+6x+9-9-7\right]}}$

$\int \frac{dx}{\sqrt{-1\left[{\left(x+3\right)}^2-16\right]}}$

$\int \frac{dx}{\sqrt{{\left(4\right)}^2-{\left(x+3\right)}^2}}$

$={\sin }^{-1}\left(\frac{x+3}{4}\right)+c$