Chapter 5.9, Problem 71ES

(a) Let

$I={\displaystyle {\int }_0^a\frac{f\left(x\right)}{f\left(x\right)+f\left(a-x\right)}dx}$

Show that $I=a/2$ .

(b) Use the result of part (a) to find

${\displaystyle {\int }_0^3\frac{\sqrt{x}}{\sqrt{x}+\sqrt{3-x}}dx}$

(c) Use the result of part (a) to find

${\displaystyle {\int }_0^{\pi /2}\frac{\sin x}{\sin x+\cos x}dx}$