Find the integral of the function sin ^{-1}(c | vedclass

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Question

(A) We know that $\sin ^{-1}(\cos x) = \sin ^{-1}(\sin(\frac{\pi}{2}-x))$.Since $\sin ^{-1}(\sin 	heta) = 	heta$ for $	heta \in [-\frac{\pi}{2}, \f

Vedclass · Accepted Answer

$\frac{\pi x}{2}-\frac{x^{2}}{2}+C_{1}$

Vedclass · Answer

(A) We know that $\sin ^{-1}(\cos x) = \sin ^{-1}(\sin(\frac{\pi}{2}-x))$.Since $\sin ^{-1}(\sin 	heta) = 	heta$ for $	heta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,we have $\sin ^{-1}(\cos x) = \frac{\pi}{2}-x$.Now,we integrate the function:$\int \sin ^{-1}(\cos x) \, dx = \int (\frac{\pi}{2}-x) \, dx$$= \int \frac{\pi}{2} \, dx - \int x \, dx$$= \frac{\pi}{2}x - \frac{x^{2}}{2} + C_{1}$.

Find the integral of the function $\sin ^{-1}(\cos x)$.

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