Chapter 4, Problem 14PS

(a)

To determine

To Prove: The specified statement, “For any continuous function on the interval $\left[0,b\right]$ with $f\left(x\right)+f\left(b-x\right)\ne 0$”.

${\displaystyle \underset{0}{\overset{b}{\int }}\frac{f\left(x\right)}{f\left(x\right)+f\left(b-x\right)}dx}=\frac{b}{2}$

(b)

To determine

To calculate: The given integral ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{\sin x}{\sin \left(1-x\right)+\sin x}dx}$ value applying the result attained in part (a).

(c)

To determine

To calculate: The value of the given integral ${\displaystyle \underset{0}{\overset{3}{\int }}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{3-x}}dx}$ using the result obtained in part (a).