\(\int \frac{2 x+5}{\sqrt{7-6 x-x^2}} \mathrm{~d} x=\mathrm{A} \sqrt{7-6 x-x^2}+\mathrm{B} \sin ^{-1}\left(\frac{x+3}{4}\right)+\mathrm{c}\) (where c c is a constant of integration) then the value of \(A+B\) is

\(\text { Let } I=\int \frac{2 x+5}{\sqrt{7-6 x-x^2}} \mathrm{~d} x=\int \frac{2 x+6-6+5}{\sqrt{7-6 x-x^2}} \)
\( =-1 \int \frac{-2 x-6}{\sqrt{7-6 x-x^2}} \mathrm{~d} x-\) \(\int \frac{1}{\sqrt{7+9-\left(9+6 x+x^2\right)}} \mathrm{d} x \)
\( =-1 \int \frac{-2 x-6}{\sqrt{7-6 x-x^2}} \mathrm{~d} x-\int \frac{1}{\sqrt{(4)^2-(x+3)^2}} \mathrm{~d} x\)
Let \(7-6 x-x^2=\mathrm{t}\)
\(\therefore (-2 x-6) \mathrm{d} x=\mathrm{dt}\)
\(\therefore \mathrm{I} =-\int(\mathrm{t})^{\frac{-1}{2}} \mathrm{dt}-\sin ^{-1}\left(\frac{x+3}{4}\right)+\mathrm{c} \)
\( =-2 \sqrt{7-6 x-x^2}-\sin ^{-1}\left(\frac{x+3}{4}\right)+\mathrm{c}\)
\(\therefore A=-2 \text { and } B=-1\)
\(\therefore A+B=-3\)