5.3 The Fundamental Theorem of Calculus/37

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{\frac {1}{2}}^{\frac {\sqrt {3}}{2}}{\frac {6}{\sqrt {1-t^{2}}}}\,dt&=6\int _{\frac {1}{2}}^{\frac {\sqrt {3}}{2}}{\frac {1}{\sqrt {1-t^{2}}}}\,dt\\[2ex]&=6\arcsin {(x)}{\bigg |}_{\frac {1}{2}}^{\frac {\sqrt {3}}{2}}\\[2ex]&=\left[6\arcsin \left({\frac {\sqrt {3}}{2}}\right)\right]-\left[6\arcsin {\left({\frac {1}{2}}\right)}\right]\\[2ex]&=\left[6\cdot {\frac {\pi }{3}}\right]-\left[6\cdot {\frac {\pi }{6}}\right]=2\pi -\pi \\[2ex]&=\pi \end{aligned}}}