Integrating arccos x

To evaluate

\[\int^1_0 cos^{-1} xdx\]

consider the following graph.

The area under the curve of

\[y=cosx\]

is adjacent to both

\[x\]

and

\[y\]

axes so

\[\int^1_0 cos^{-1} y dy= \int^{\pi /2}_0 cosx dx= [sinx]^{\pi /2}_0 =1\]

.