If $\int \sin \left(101\mathrm{x}\right)(\mathrm{sinx}{)}^{99}\mathrm{dx}=\frac{\sin \left(100\mathrm{x}\right)(\mathrm{sinx}{)}^{\mathrm{\lambda }}}{\mathrm{\mu }}+\mathrm{c}$ then $\frac{\mathrm{\lambda }}{\mathrm{\mu }}=$

$=\int \sin \left(101x\right)\cdot (\sin x{)}^{99}\cdot dx$

$=\int \sin (100x+x)\cdot (\sin x{)}^{99}\cdot dx$

$=\int (\sin (100x)\cdot \cos x+\cos (100x)\cdot \sin x)(\sin x{)}^{99}\cdot dx$

$=\int \sin \left(100x\right)\cdot (\sin x{)}^{99}\cdot \cos x\cdot dx+\int \cos (100x)\cdot (\sin x{)}^{100}\cdot dx$

$=\frac{1}{100}\sin \left(100x\right)(\sin x{)}^{100}-\int \cos (100x)\cdot (\sin x{)}^{100}\cdot dx+\int \cos \left(100x\right)\cdot (\sin x{)}^{100}\cdot dx$

$=\frac{1}{100}\sin \left(100x\right)\cdot (\sin x{)}^{100}+C$