5.3 The Fundamental Theorem of Calculus/37: Difference between revisions
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FTC # 2- the <math>\frac{d}{dx}[\int\limits_{a(x)}^{b(x)}F(x)dx]</math> is F(b)-F(a) where F is the antiderivitive of f such that <math>F^\prime=f</math> | FTC # 2- the <math>\frac{d}{dx}[\int\limits_{a(x)}^{b(x)}F(x)dx]</math> is F(b)-F(a) where F is the antiderivitive of f such that <math>F^\prime=f</math> | ||
37) <math>\int\limits_{1/2}^{\sqrt{3})/2}\frac{6}{\sqrt{1-t^2}} dt </math> | 37) g(x)=<math>\int\limits_{1/2}^{\sqrt{3})/2}\frac{6}{\sqrt{1-t^2}} dt </math> | ||
<math>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}</math> | |||
<math>6sin^{-1}(x)\bigg|_{1/2}^{\sqrt{3}/2}</math> | |||
=<math>6sin^{-1}(\sqrt{3})/2)-(6sin^{-1}(1/2))</math> | =<math>6sin^{-1}(\sqrt{3})/2)-(6sin^{-1}(1/2))</math> | ||
![{\displaystyle {\frac {d}{dx}}[\int \limits _{a(x)}^{b(x)}F(x)dx]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ba58f09e91a121dd850cf63fc93a0c2b6af88084)
