Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 191049 by Mingma last updated on 16/Apr/23

Commented by Mingma last updated on 17/Apr/23

Nice solution, sir!

Answered by aleks041103 last updated on 17/Apr/23

$$cos\left(sin\left(x\right)\right)=Re\left(exp\left(isin\left(x\right)\right)\right)$$$$\Rightarrow cos\left(sin\left(x\right)\right)exp\left(cos\left(x\right)\right)=$$$$=Re\left(exp(cos\left(x\right)+isin\left(x\right))\right)=$$$$=Re\left(e^{e^{ix}}\right)$$$$\Rightarrow {\int }_0^{2\pi }cos\left(sin\left(x\right)\right)exp\left(cos\left(x\right)\right)dx=$$$$=Re\left({\int }_0^{2\pi }{e}^{e^{ix}}dx\right)$$$${\int }_0^{2\pi }{e}^{e^{ix}}dx=?$$$$z=e^{ix}\Rightarrow dz=izdx\Rightarrow dx=-i\frac{dz}{z}$$$$\Rightarrow {\int }_0^{2\pi }{e}^{e^{ix}}dx=-i{\oint }_{\mathrm{\Gamma }}\frac{e^z}{z}dz$$$$where\mathrm{\Gamma }:\mid z\mid =1$$$$Byresiduetheorem:$$$${\oint }_{\mathrm{\Gamma }}\frac{e^z}{z}dz=2\pi iRes(z=0)$$$$Res(z=0)=\underset{z\to 0}{lim}\frac{e^z}{z}(z-0)=1\Rightarrow$$$$\Rightarrow {\int }_0^{2\pi }{e}^{e^{ix}}dx=-i\left(2\pi i\right)=2\pi$$$$\Rightarrow {\int }_0^{2\pi }cos\left(sin\left(x\right)\right)exp\left(cos\left(x\right)\right)dx=2\pi$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com