Question

Evaluate:$\int \frac{1+3x}{\sqrt{5-2x-x^2}}dx$

Answer 1

$Let1+3x=\left(A\frac{d}{dx}\right[5-2x-x^2]+B\Rightarrow 1+3x=A(-2x-2)+B\Rightarrow A=-\frac{3}{2},B=-2$

$\int \frac{1+3x}{\sqrt{5-2x-x^2}}dx=-\frac{3}{2}\int \frac{-2-2x}{\sqrt{5-2x-x^2}}dx-2\int \frac{1}{\sqrt{5-2x-x^2}}dx$

$\Rightarrow =-\frac{3}{2}\left[2\sqrt{5-2x-x^2}\right]-2\int \frac{dx}{\sqrt{{\left[\sqrt{6}\right]}^2-{[x+1]}^2}}$

$\Rightarrow =-3\sqrt{5-2x-x^2}-2si{n}^{-1}\left(\frac{x+1}{\sqrt{6}}\right)+C$