Question

\[\int_0^{2\pi} \cos^{- 1} \left( \cos x \right)dx\]

Answer 1

\[\int_0^{2\pi} \cos^{- 1} \left( \cos x \right)dx\]

\[ = \int_0^\pi \cos^{- 1} \left( \cos x \right)dx +\int_\pi^{2\pi} \cos^{- 1} \left( \cos x \right)dx\]

\[ = \int_0^\pi xdx + \int_\pi^{2\pi} \left( 2\pi - x \right)dx .....................\left[ \pi \leq x \leq 2\pi \Rightarrow - 2\pi \leq - x \leq - \pi \Rightarrow 0 \leq 2\pi - x \leq \pi \right]\]

\[= \left.\frac{x^2}{2}\right|_0^\pi + \left.\frac{\left( 2\pi - x \right)^2}{2 \times \left( - 1 \right)}\right|_\pi^{2\pi} \]
\[ = \frac{1}{2}\left( \pi^2 - 0 \right) - \frac{1}{2}\left( 0 - \pi^2 \right)\]
\[ = \frac{\pi^2}{2} + \frac{\pi^2}{2}\]
\[ = \pi^2\]