Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 199719 by sonukgindia last updated on 08/Nov/23

Answered by witcher3 last updated on 08/Nov/23

$${\mathrm{I}}_{\mathrm{a}}={\mathrm{I}}_{\mathrm{b}}$$$${\mathrm{I}}_{\mathrm{a}}={\int }_0^{\pi }{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}+{\int }_{\pi }^{2\pi }{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}(\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}$$$$={\int }_0^{\pi }{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}(\cos \left(\cos \left(\mathrm{x}\right)\right)+{\mathrm{e}}^{-\sin \left(\mathrm{x}\right)}(\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}$$$${\mathrm{I}}_{\mathrm{b}}={\int }_0^{\pi }{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}(\cos \left(\sin \left(\mathrm{x}\right)\right)+{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}(\cos \left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$$$$={\int }_0^{\frac{\pi }{2}}({\mathrm{e}}^{\cos \left(\mathrm{x}\right)}(\cos \left(\sin \left(\mathrm{x}\right)\right)+{\mathrm{e}}^{\cos \left(\mathrm{x}\right)}(\cos \left(\sin \left(\mathrm{x}\right)\right){\mathrm{dx}}^{\prime }\mathrm{x}\to \frac{\pi }{2}-\mathrm{x}$$$$+{\int }_{\frac{\pi }{2}}^{\pi }(........)..\mathrm{x}\to \frac{\pi }{2}+\mathrm{x}$$$$=2{\int }_0^{\frac{\pi }{2}}{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}+2{\int }_0^{\frac{\pi }{2}}{\mathrm{e}}^{-\sin \left(\mathrm{x}\right)}(\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}$$$$={\int }_0^{\pi }{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\cos \left(\cos \left(\mathrm{x}\right)\right)+{\mathrm{e}}^{-\sin \left(\mathrm{x}\right)}\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}={\mathrm{I}}_{\mathrm{b}}$$$${\mathrm{I}}_{\mathrm{a}}=2{\int }_0^{\pi }(\cos (\cos \left(\mathrm{x}\right)\mathrm{ch}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$$$$=4{\int }_0^{\frac{\pi }{2}}(\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{ch}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$$$$=4{\int }_0^1\cos \left(\cos \left(\mathrm{x}\right)\right)\sum _{\mathrm{n}⩾0}\frac{{\sin }^{2\mathrm{n}}\left(\mathrm{x}\right)}{\left(2\mathrm{n}\right)!}\mathrm{dx}$$$$=4{\int }_0^{\frac{\pi }{2}}\cos \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx}+4\sum _{\mathrm{n}⩾1}{\int }_0^{\frac{\pi }{2}}\cos \left(\cos \left(\mathrm{x}\right)\right)\sum _{\mathrm{n}⩾1}\frac{{\sin }^{2\mathrm{n}}\left(\mathrm{x}\right)}{\left(2\mathrm{n}\right)!}\mathrm{dx}$$$$4\mathrm{A}+4\mathrm{B};\mathrm{A}=2\pi {\mathrm{J}}_0\left(1\right).\mathrm{easy}$$$$\mathrm{B}=\sum _{\mathrm{n}⩾1}{\int }_0^1\cos \left(\mathrm{t}\right){\left(\sqrt{1-{\mathrm{t}}^2}\right)}^{2\mathrm{n}-1}\mathrm{dt}$$$$=\sum _{\mathrm{n}⩾1}{\int }_0^1\cos \left(\mathrm{t}\right){(1-{\mathrm{t}}^2)}^{\mathrm{n}-\frac{1}{2}}\mathrm{dt}$$$${\int }_0^1\cos \left(\mathrm{t}\right){(1-{\mathrm{t}}^2)}^{\mathrm{a}}\mathrm{dt}=\mathrm{E}$$$$=\sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}}{\left(2\mathrm{n}\right)!}{\int }_0^1{\mathrm{t}}^{2\mathrm{n}}{(1-{\mathrm{t}}^2)}^{\mathrm{a}}\mathrm{dt}$$$$\mathrm{y}={\mathrm{t}}^2\iff \sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}}{\left(2\mathrm{n}\right)!}\frac{1}{2}{\int }_0^1{\mathrm{y}}^{\mathrm{n}-\frac{1}{2}}{(1-\mathrm{y})}^{\mathrm{a}}\mathrm{dy}$$$$=\sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}}{2\left(2\mathrm{n}\right)!}\frac{\mathrm{\Gamma }(\mathrm{n}+\frac{1}{2})\mathrm{\Gamma }(1+\mathrm{a})}{\mathrm{\Gamma }(\mathrm{n}+\mathrm{a}+\frac{3}{2})}$$$${\mathrm{J}}_{\mathrm{m}}\left(\mathrm{z}\right)\sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}{\mathrm{z}}^{2\mathrm{n}+\mathrm{m}}}{2^{2\mathrm{n}+\mathrm{m}}\left(\mathrm{n}\right)!(\mathrm{n}+\mathrm{m})!};\mathrm{bassel}\mathrm{function}$$$$=\frac{\mathrm{\Gamma }(1+\mathrm{a})}{2}\sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}\mathrm{\Gamma }(\mathrm{n}+\frac{1}{2})}{(\mathrm{n}+\mathrm{a}+\frac{1}{2})!.\left(2\mathrm{n}\right)!}...$$$$\left(2\mathrm{n}\right)!=2^{\mathrm{n}}.\mathrm{n}!.2^{\mathrm{n}}\underset{\mathrm{k}=0}{\prod ^{\mathrm{n}-1}}(\mathrm{k}+\frac{1}{2})$$$$=2^{2\mathrm{n}}.\mathrm{n}!.\frac{\mathrm{\Gamma }(\mathrm{n}+\frac{1}{2})}{\mathrm{\Gamma }\left(\frac{1}{2}\right)}$$$$\mathrm{E}=\frac{\mathrm{\Gamma }(1+\mathrm{a})2^{\mathrm{a}+\frac{1}{2}}}{2}.\sqrt{\pi }\sum _{\mathrm{n}⩾0}\frac{{(-1)}^{\mathrm{n}}}{2^{2\mathrm{n}+\mathrm{a}+\frac{1}{2}}.\mathrm{n}!.(\mathrm{n}+\mathrm{a}+\frac{1}{2})}$$$$=2^{\mathrm{a}-\frac{1}{2}}\sqrt{\pi }\mathrm{\Gamma }(1+\mathrm{a}){\mathrm{J}}_{\mathrm{a}+\frac{1}{2}}\left(1\right)$$$$\mathrm{B}=\sum _{\mathrm{n}⩾1}{2}^{\mathrm{n}-1}\sqrt{\pi }\mathrm{\Gamma }(1+\frac{\mathrm{n}}{2}){\mathrm{J}}_{\mathrm{n}}\left(1\right)$$$${\mathrm{I}}_{\mathrm{a}}=2\pi {\mathrm{J}}_0\left(1\right)+\sum _{\mathrm{n}⩾1}{2}^{\mathrm{n}-1}\sqrt{\pi }\mathrm{\Gamma }(1+\frac{\mathrm{n}}{2}){\mathrm{J}}_{\mathrm{n}}\left(1\right)$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com