Complementary inverses

Calculus Level 4

Evaluate $\displaystyle \int_{1729 \pi}^{1729.5 \pi} \left[ \sin^{-1} (\cos x) +\cos^{-1} ( \sin x ) \right] dx$ .

\frac{\pi^2}{2}

\frac{\pi^2}{4}

\frac{\pi^2}{8}

\pi^2

1 solution

Vincent Moroney
Jun 14, 2018

$\begin{aligned} I = & \int_{1729\pi}^{1729.5\pi} \Big[ \arcsin(\cos x) + \arccos(\sin x) \Big]\, dx\\ = & \int_{0}^{\frac{\pi}{2}} \Big[ \arcsin(\cos x) + \arccos(\sin x) \Big]\, dx\\ = & \int_0^{\frac{\pi}{2}} \Big[ \arcsin(\sin(x+\frac{\pi}{2})) + \arccos(\cos(x-\frac{\pi}{2})) \Big]\, dx\\ = & \int_0^{\frac{\pi}{2}} 2x\, dx\\ = & \boxed{\frac{\pi^2}{4}} \end{aligned}$

Complementary inverses

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