Question

$\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{3}}}{\mathbf{\int }}}\frac{\mathbf{1}}{\sqrt{\mathbf{9}\mathbf{-}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}}\mathit{d}\mathit{\theta }\mathbf{.}$

Short answer

The value of integral in elliptic form is $\frac{1}{3}F\left(\frac{\mathrm{\pi }}{3},\frac{1}{3}\right)\approx 0.35499$.

Step-by-step solution

Given Information

The value of integration is $\underset{0}{\overset{\frac{\mathrm{\pi }}{3}}{\int }}\frac{1}{\sqrt{9-{\sin }^2\theta }}d\theta$.

Definition of elliptic form.

The elliptic form of the integral is defined as $\mathit{F}\left(\frac{\pi }{2},k\right)\mathbf{=}\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{2}}}{\mathbf{\int }}}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{k}}^{\mathbf{2}}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}}\mathit{d}\mathit{\theta }$.

Find the value of Integral.

The value of integration is $\underset{0}{\overset{\frac{\mathrm{\pi }}{3}}{\int }}\frac{1}{\sqrt{9-{\sin }^2\theta }}d\theta$.Factor out 9 the equation becomes as follows.

$$\begin{gathered} l=\underset{0}{\overset{\frac{\mathrm{\pi }}{3}}{\int }}\frac{1}{\sqrt{9\left(1-{\displaystyle \frac{{\sin }^2\theta }{9}}\right)}}d\theta \\ l=\underset{0}{\overset{\frac{\mathrm{\pi }}{3}}{\int }}\frac{1}{3\sqrt{\left(1-{\displaystyle \frac{{\sin }^2\theta }{9}}\right)}}d\theta \end{gathered}$$

The formula for the beta function is $F(\frac{\mathrm{\pi }}{2},\mathrm{k})=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{1}{\sqrt{1-{\mathrm{k}}^2{\sin }^2\mathrm{\theta }}}d\theta$.

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

$$\begin{gathered} l=\frac{1}{3}\underset{0}{\overset{\frac{\mathrm{\pi }}{3}}{\int }}\frac{1}{\sqrt{\left(1-{\displaystyle \frac{{\sin }^2\theta }{9}}\right)}}d\theta \\ l=\frac{1}{3}F\left(\frac{\mathrm{\pi }}{3},\frac{1}{3}\right) \\ l\approx 0.35499 \end{gathered}$$

Hence, The value of integral in elliptic form is $\frac{1}{3}F\left(\frac{\mathrm{\pi }}{3},\frac{1}{3}\right)\approx 0.35499$.