5.3 The Fundamental Theorem of Calculus/37

FTC # 2- the ${\frac {d}{dx}}\left(\int \limits _{a(x)}^{b(x)}F(x)dx\right)$ is F(b)-F(a) where F is the antiderivitive of f such that $F^{\prime }=f$

37) g(x)=$\int \limits _{1/2}^{{\sqrt {3}}/2}{\frac {6}{\sqrt {1-t^{2}}}}dt$

$g^{\prime }(x)={\frac {d}{dx}}\left(\int \limits _{1/2}^{{\sqrt {3}}/2}{\frac {6}{\sqrt {1-t^{2}}}}dt\right)=6sin^{-1}(x){\bigg |}_{1/2}^{{\sqrt {3}}/2}$

=$6sin^{-1}({\sqrt {3}})/2)-(6sin^{-1}(1/2))$

=$\pi$

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