If $\int \frac{x+3}{\sqrt{5-4 x+x^2}} d x=A \int \frac{(-4-2 x)}{\sqrt{5-4 x+x^2}} d x+B \int \frac{

Question

If $\int \frac{x+3}{\sqrt{5-4 x+x^2}} d x=A \int \frac{(-4-2 x)}{\sqrt{5-4 x+x^2}} d x+B \int \frac{d x}{\sqrt{5-4 x+x^2}}$; then $2 A+B$ is

Professor PC Thomas Classes · Answer

Equate numerators: $x+3 = A(-4-2x) + B \implies -2A = 1 \implies A = -1/2$, and $-4A+B = 3 \implies 2+B=3 \implies B=1$. Thus $2A+B = -1+1=0$.