integration of composition of 2 functions

Question

$$\int_{\cos3}^1\lceil\arccos x\rceil\,dx$$

I tried to break the integral into parts by looking the values from graph , like from $\cos3$ to $\cos2$ etc. But, i dont get the answer.

Answer 1

By splitting, we have that \begin{align*} \int_{\cos3}^1\lceil\arccos x\rceil\,dx &=\int_{\cos3}^{\cos 2}\lceil\arccos x\rceil\,dx+\int_{\cos 2}^{\cos 1}\lceil\arccos x\rceil\,dx+\int_{\cos 1}^{\cos 0}\lceil\arccos x\rceil\,dx= \\ &=\int_{\cos3}^{\cos 2}3\,dx+\int_{\cos 2}^{\cos 1}2\,dx+\int_{\cos 1}^{\cos 0}1\,dx\\ &=3(\cos 2-\cos 3)+2(\cos 1-\cos 2)-(\cos 1-1)\\ &=-3\cos 3+\cos 2 +\cos 1+1. \end{align*}