Let the domain of the function f(x)=_4(_5(_3(18x-x^2-77))) be (a,b).

Question

Let the domain of the function $f(x)=\log_4(\log_5(\log_3(18x-x^2-77)))$ be $(a,b)$. Then the value of the integral $\displaystyle\int_a^b\frac{\sin^3x}{\sin^3x+\sin^3(a+b-x)}\,dx$ is equal to \_\_\_\_\_\_\_\_.

Answer: 1

Accepted Answer

For the domain, we need $\log_5(\log_3(18x-x^2-77))>0\Rightarrow\log_3(18x-x^2-77)>1\Rightarrow18x-x^2-77>3$
$$\Rightarrow x^2-18x+80<0\Rightarrow x\in(8,10)\Rightarrow a=8,\ b=10$$

Let $I=\displaystyle\int_a^b\frac{\sin^3x}{\sin^3x+\sin^3(a+b-x)}dx$. Using $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$:
$$I=\int_a^b\frac{\sin^3(a+b-x)}{\sin^3x+\sin^3(a+b-x)}dx$$

Adding both forms: $2I=(b-a)\Rightarrow I=\dfrac{b-a}{2}=\dfrac{10-8}{2}=1$

Solutions · 1