Mathematics - Functions Question with Solution | TestHub

For domain ${\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)>0$

${\log }_3\left(18x-x^2-77\right)>1$

$\Rightarrow 18x-x^2-77>3$

$\Rightarrow x^2-18x+80<0$

$\Rightarrow x\in \left(8,10\right)$

$\Rightarrow a=8$ and $b=10$

$I={\int }_a^b\frac{{\sin }^{3}x}{{\sin }^{3}x+{\sin }^3\left(a+b-x\right)}dx$

Apply ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f\left(a+b-x\right)dx$

$I={\int }_a^b\frac{{\sin }^3\left(a+b-x\right)}{{\sin }^{3}x+{\sin }^3\left(a+b-x\right)}dx$

$\Rightarrow 2I=\left(b-a\right)\Rightarrow I=\frac{b-a}{2}\left(\because a=8 \& b=10\right)$

$\Rightarrow I=\frac{10-8}{2}=1$