Differentiation Questions

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

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{dx}}{\left(\sqrt{\mathrm{5}}−\mathrm{cos}\:{x}\right)^{\mathrm{3}} }\:? \\$$

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

$${F}\left({a}\right)\:=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{dx}}{{a}−\mathrm{cos}\:{x}}\:=\:\frac{\pi}{\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}} \\$$$${F}\:'\left({a}\right)\:=\:\int\:\frac{\partial}{\partial{a}}\left[\frac{{dx}}{{a}−\mathrm{cos}\:{x}}\:\right]\:=\:\frac{\partial}{\partial{a}}\:\left[\frac{\pi}{\sqrt{{a}^{\mathrm{2}} −\mathrm{1}}}\right] \\$$$$−\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\left(\frac{{dx}}{\left({a}−\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\right)=−\pi{a}^{\mathrm{2}} \left({a}^{\mathrm{2}} −\mathrm{1}\right)^{−\mathrm{3}/\mathrm{2}} \\$$$${F}\:''\left({a}\right)=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\partial}{\partial{a}}\left[\frac{{dx}}{\left({a}−\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\right]=−\pi\left[\left({a}^{\mathrm{2}} −\mathrm{1}\right)^{−\mathrm{3}/\mathrm{2}} −\mathrm{3}{a}^{\mathrm{2}} \left({a}^{\mathrm{2}} −\mathrm{1}\right)^{−\mathrm{5}/\mathrm{2}} \right] \\$$$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{dx}}{\left(\mathrm{1}−\mathrm{cos}\:{x}\right)^{\mathrm{3}} }\:=\:−\frac{\pi}{\mathrm{2}}\left({a}^{\mathrm{2}} −\mathrm{1}\right)^{−\mathrm{5}/\mathrm{2}} \left(\left({a}^{\mathrm{2}} −\mathrm{1}\right)−\mathrm{3}{a}^{\mathrm{2}} \right) \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\pi}{\mathrm{2}}.\:\frac{\mathrm{2}{a}^{\mathrm{2}} +\mathrm{1}}{\left({a}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}/\mathrm{2}} } \\$$$${put}\:{a}\:=\:\sqrt{\mathrm{5}} \\$$$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{dx}}{\left(\sqrt{\mathrm{5}}−\mathrm{cos}\:{x}\right)^{\mathrm{3}} }\:=\:\frac{\mathrm{11}\pi}{\mathrm{64}}\:.\:\bigstar \\$$