Question

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

9. $\underset{\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}}}{\overset{\frac{\mathbf{1}}{\mathbf{2}}}{\mathbf{\int }}}\frac{\mathbf{d}\mathbf{t}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\sqrt{\mathbf{4}\mathbf{-}\mathbf{3}{\mathbf{t}}^{\mathbf{2}}}}$.

Short answer

The value of integral in elliptic form is $F\left({\sin }^{-1}\left(\frac{1}{2}\right),\frac{\sqrt{3}}{4}\right)\approx 0.26696$.

Step-by-step solution

Given Information

The given integral is $\underset{\frac{-1}{2}}{\overset{\frac{1}{2}}{\int }}\frac{1}{\sqrt{1-t^2}\sqrt{4-3t^2}}dt$.

Definition of elliptic form

The elliptic form of the integral is defined as $\mathit{F}\left(\varphi ,k\right)\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{k}}^{\mathbf{2}}{\mathbf{t}}^{\mathbf{2}}}}\mathit{d}\mathit{t}$, where $\mathit{x}\mathbf{=}\mathbf{sin}\mathit{\varphi }$.

Find the value of the Integral

Let the given integral is $l=\underset{\frac{-1}{2}}{\overset{\frac{1}{2}}{\int }}\frac{1}{\sqrt{1-t^2}\sqrt{4-3t^2}}dt$.

Factor out 4 in the denominator, then the equation becomes as follows:

$l=\underset{\frac{-1}{2}}{\overset{\frac{1}{2}}{\int }}\frac{1}{\sqrt{1-t^2}\sqrt{1-{\displaystyle \frac{3}{4}}{t}^2}}dt$

The formula for the beta function is $F\left(\varphi ,k\right)=\underset{0}{\overset{x}{\int }}\frac{1}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}dt$.

Equate the above equation with the value of I, and the value of I becomes follows.

localid="1664971098445" $\begin{array}{rcl}l& =& \frac{1}{2}\underset{\frac{-1}{2}}{\overset{\frac{1}{2}}{\int }}\frac{1}{\sqrt{1-t^2}\sqrt{1-{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2{t}^2}}dt\\ l& =& \frac{1}{2}F\left({\sin }^{-1}\left(\frac{1}{2}\right),\frac{\sqrt{3}}{4}\right)\\ l& \approx & \frac{1}{2}\times 0.53392\\ & \approx & 0.26696\end{array}$

The value of integral in elliptic form is $\begin{array}{rcl}F\left({\sin }^{-1}\left(\frac{1}{2}\right),\frac{\sqrt{3}}{4}\right)& \approx & 0.26696\end{array}$.