5.3 The Fundamental Theorem of Calculus/37: Difference between revisions

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<math>
<math>int\limits_{1/2}^{frac{root^3}{(2)}\frac{1}{(1+u^2)}u^3, du </math>
\begin{align}


[[5.3 The Fundamental Theorem of Calculus/1|1]]
\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]
[[5.3 The Fundamental Theorem of Calculus/3|3]]
 
[[5.3 The Fundamental Theorem of Calculus/5|5]]
&=6\arcsin{(x)}\bigg|_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \\[2ex]
[[5.3 The Fundamental Theorem of Calculus/7|7]]
 
[[5.3 The Fundamental Theorem of Calculus/8|8]]
&=\left[6\arcsin\left(\frac{\sqrt{3}}{2}\right)\right]-\left[6\arcsin{\left(\frac{1}{2}\right)}\right] \\[2ex]
[[5.3 The Fundamental Theorem of Calculus/9|9]]
 
[[5.3 The Fundamental Theorem of Calculus/10|10]]
&=\left[6\cdot\frac{\pi}{3}\right]-\left[6\cdot\frac{\pi}{6}\right] = 2\pi-\pi \\[2ex]
[[5.3 The Fundamental Theorem of Calculus/11|11]]
 
[[5.3 The Fundamental Theorem of Calculus/13|13]]
&=\pi
[[5.3 The Fundamental Theorem of Calculus/15|15]]
 
[[5.3 The Fundamental Theorem of Calculus/17|17]]
\end{align}
[[5.3 The Fundamental Theorem of Calculus/19|19]]
</math>
[[5.3 The Fundamental Theorem of Calculus/20|20]]
[[5.3 The Fundamental Theorem of Calculus/21|21]]
[[5.3 The Fundamental Theorem of Calculus/23|23]]
[[5.3 The Fundamental Theorem of Calculus/25|25]]
[[5.3 The Fundamental Theorem of Calculus/27|27]]
[[5.3 The Fundamental Theorem of Calculus/28|28]]
[[5.3 The Fundamental Theorem of Calculus/29|29]]
[[5.3 The Fundamental Theorem of Calculus/31|31]]
[[5.3 The Fundamental Theorem of Calculus/33|33]]
[[5.3 The Fundamental Theorem of Calculus/35|35]]
[[5.3 The Fundamental Theorem of Calculus/37|37]]
[[5.3 The Fundamental Theorem of Calculus/39|39]]
[[5.3 The Fundamental Theorem of Calculus/41|41]]
[[5.3 The Fundamental Theorem of Calculus/53|53]]

Latest revision as of 21:59, 6 September 2022

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }

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