${\int }_{ }^{ }\frac{\cos 9\mathrm{x}+\cos 6\mathrm{x}}{-1+2\cos 5\mathrm{x}}\mathrm{dx}=\mathrm{Asin}4\mathrm{x}+\mathrm{Bsinx}+\mathrm{C}$ where ‘C’ is a constant of integration, than A + B =

$\begin{array}{l}\int \frac{2\cos \left(\frac{15\mathrm{x}}{2}\right)\cos \left(\frac{3\mathrm{x}}{2}\right)}{2\left(2{\cos }^2\frac{5\mathrm{x}}{2}-1\right)-1}\mathrm{dx}\\ =\int \frac{2\cos \left(\frac{15\mathrm{x}}{2}\right)\cos \left(\frac{3\mathrm{x}}{2}\right)\cos \left(\frac{5\mathrm{x}}{2}\right)}{\left(4{\cos }^2\left(\frac{5\mathrm{x}}{2}\right)-3\right)\cos \left(\frac{5\mathrm{x}}{2}\right)}\mathrm{dx}\\ =\int \frac{2\cos \left(\frac{15\mathrm{x}}{2}\right)\cos \left(\frac{3\mathrm{x}}{2}\right)\cos \left(\frac{5\mathrm{x}}{2}\right)}{4{\cos }^3\left(\frac{5\mathrm{x}}{2}\right)-3\cos \left(\frac{5\mathrm{x}}{2}\right)}\mathrm{dx}\\ =\int \frac{2\cos \left(\frac{15\mathrm{x}}{2}\right)\cos \left(\frac{3\mathrm{x}}{2}\right)\cos \left(\frac{15\mathrm{x}}{2}\right)}{\cos \left(\frac{15\mathrm{x}}{2}\right)}\mathrm{dx}\\ =\int (\cos 4\mathrm{x}+\cos \mathrm{x})\mathrm{dx}=\frac{1}{4}\sin 4\mathrm{x}+\sin \mathrm{x}+\mathrm{C}\end{array}$