Question

If $-{\int }_0^{200\pi }[\sin x+\cos x]dx=A\pi$ and $\left[x\right]$ denotes the greatest integer function then $A$ is equal to

Answer 1

Let $f\left(x\right)=\sin x+\cos x$ By considering the minimum and maximum value of $f\left(x\right)$ on $[0,\pi /2],[\pi /2,3\pi /4],...[7\pi /4,2\pi ]$
$\left[f\right(x\left)\right]=\{\begin{array}{cc}1& 0\le x\le \pi /2\\ 0& \pi /2<x\le 3\pi /4\\ -1& 3\pi /4<x\le \pi \\ -2& \pi <x<3\pi /4\\ -1& 3\pi /2\le x<7\pi /4\\ 0& 7\pi /4\le x<2\pi \end{array}$
so, ${\int }_0^{2\pi }[\sin x+\cos x]dx={\int }_0^{\pi /2}1dx+{\int }_{\pi /2}^{3\pi /4}0dx$ $+{\int }_{3\pi /4}^{\pi }(-1)dx+{\int }_{\pi }^{3\pi /2}(-2)dx+{\int }_{3\pi /2}^{7\pi /4}(-1)dx+{\int }_{7\pi /4}^{2\pi }0dx$
$=\frac{\pi }{2}+\frac{3\pi }{4}-\pi +2\pi -3\pi +\frac{3\pi }{2}-\frac{7\pi }{4}=-\pi$
Since $\sin x+\cos x$ is a periodic function with period $2\pi ,{\int }_0^{200\pi }[\sin x+\cos x]dx=-100\pi$