( $\int_{- 1}^{1} [{cos}^{- 1} x] d x$ where $[.]$ denotes greatest integer function)

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Solution

REF . Image.
Thus,
$\int_{- 1}^{1} [c o s^{- 1} x] d x$
$= \int_{- 1}^{c o s 3} 3 d x + \int_{c o s 3}^{c o s 2} 2 d x + \int_{c o s 2}^{c o s 1} 1 d x + \int_{c o s 1}^{1} 0 d x$
$= 3 [c o s 3 + 1] + 2 [c o s 2 - c o s 3] + c o s 1 - c o s 2 + 0$
$= 3 + c o s 3 + c o s 2 + c o s 1$

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