Integral

[Filip84]

\(\displaystyle \phi\left(\theta\right)\, =\, \displaystyle{\int\, \frac{k\csc^2\left(\theta\right)}{\sqrt{(1\, -\, k^2)\, -\, \left(k \cot\left(\theta\right)\right)^2}}\, d\theta}\, +\, c_2\)

Can somebody explain how to solve integral above? The solution is:

. . . . .\(\displaystyle \phi\left(\theta\right)\, =\, \cos^{-1} \displaystyle{ \left\{ \frac{k \cot \left( \theta \right) } { \sqrt{1\, -\, k^2} } \right\}} \, +\, c_2\)

 

[Deleted member 4993]

\(\displaystyle \phi\left(\theta\right)\, =\, \displaystyle{\int\, \frac{k\csc^2\left(\theta\right)}{\sqrt{(1\, -\, k^2)\, -\, \left(k \cot\left(\theta\right)\right)^2}}\, d\theta}\, +\, c_2\)

Can somebody explain how to solve integral above? The solution is:

. . . . .\(\displaystyle \phi\left(\theta\right)\, =\, \cos^{-1} \displaystyle{ \left\{ \frac{k \cot \left( \theta \right) } { \sqrt{1\, -\, k^2} } \right\}} \, +\, c_2\)

substitute

k*cot(Θ) = √(1-k²) * u ............edit

and continue.....