Question

Find the value of
$\int \frac{5x+7}{\sqrt{3-2x-x^2}}dx$

Answer 1

Let $(5x+7)=A(-2-2x)+B......\left(1\right)$

On comparing coefficients of line terms

$5=-2A$

$A=\frac{5}{-2}$

$7=-2A+B$

$7=5+B$

$B=2$

On putting value of $A$ and $B$ in equation (1)

$5x+7=-\frac{5}{2}(-2-2x)+2$

$I=\int \frac{-\frac{5}{2}(-2-2x)+2}{\sqrt{3-2x-x^2}}$

$=\int \frac{-\frac{5}{2}dt}{\sqrt{t}}+\int \frac{2}{\sqrt{3+1-1-2x-x^2}}dx=-\frac{5}{2}2\sqrt{t}+\int \frac{2}{\sqrt{4-{(x+1)}^2}}dx$

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

$=-5\sqrt{3-2x-x^2}+2{\sin }^{-1}\left(\frac{x+1}{2}\right)+c$