Question

Evaluate:

\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]

Answer 1

\[\int \cos^{-1} \left(\sin x \right)\text{ dx }\]

\[= \int \cos^{-1} \left( \cos\left( \frac{\pi}{2} - x \right) \right) \text{ dx }\]

\[ = \int \left(\frac{\pi}{2} - x \right) dx\]

\[ = \frac{\pi}{2}x - \frac{1}{2} x^2 + c\]

\[\text{Hence,}\int \cos^{-1} \left(\sin x \right)\text{ dx } = \frac{\pi}{2}x - \frac{1}{2} x^2 + c\]