Chapter 5, Problem 2PS

(a)

To determine

To graph: The function $y=\sin x$ on the interval $\left[0,2\pi \right]$ and use symmetry to conclude that ${\displaystyle {\int }_0^{2\pi }\sin xdx}=0$.

(b)

To determine

To graph: The function $y=\sin x+2$ on the interval $\left[0,2\pi \right]$ and use symmetry to evaluate ${\displaystyle {\int }_0^{2\pi }\sin xdx}$.

(c)

To determine

To graph: The function $y=\arccos x$ on the interval $\left[-1,1\right]$ and use symmetry to evaluate ${\displaystyle {\int }_{-1}^1\arccos xdx}$.

(d)

To determine

To calculate: The value of the integral ${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{1+{\left(\tan x\right)}^{\sqrt{2}}}dx}$.