0-pi-dx-5-cos-x-3-

Question Number 105706 by bramlex last updated on 31/Jul/20

\underset{0}{\int^{π}} \frac{d x}{{(\sqrt{5} - \cos x)}^{3}} ?

Answered by john santu last updated on 31/Jul/20

F (a) = \underset{0}{\int^{π}} \frac{d x}{a - \cos x} = \frac{π}{\sqrt{a^{2} - 1}}

F^{'} (a) = \int \frac{\partial}{\partial a} [\frac{d x}{a - \cos x}] = \frac{\partial}{\partial a} [\frac{π}{\sqrt{a^{2} - 1}}]

- \underset{0}{\int^{π}} (\frac{d x}{{(a - \cos x)}^{2}}) = - π a^{2} {(a^{2} - 1)}^{- 3 / 2}

F^{″} (a) = \underset{0}{\int^{π}} \frac{\partial}{\partial a} [\frac{d x}{{(a - \cos x)}^{2}}] = - π [{(a^{2} - 1)}^{- 3 / 2} - 3 a^{2} {(a^{2} - 1)}^{- 5 / 2}]

\underset{0}{\int^{π}} \frac{d x}{{(1 - \cos x)}^{3}} = - \frac{π}{2} {(a^{2} - 1)}^{- 5 / 2} ((a^{2} - 1) - 3 a^{2})

= \frac{π}{2} . \frac{2 a^{2} + 1}{{(a^{2} - 1)}^{5 / 2}}

p u t a = \sqrt{5}

\underset{0}{\int^{π}} \frac{d x}{{(\sqrt{5} - \cos x)}^{3}} = \frac{11 π}{64} . ★