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