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 7397 by Tawakalitu. last updated on 26/Aug/16

Answered by sandy_suhendra last updated on 27/Aug/16

sin x = Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)!)) (1/x) sin x = (1/x) Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)!)) = Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2n+1)!)) ∫ ((sin x)/x) dx = ∫ Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2n+1)!)) dx = Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)(2n+1)!)) + c

$${sin}\:{x}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \:{x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$$\frac{\mathrm{1}}{{x}}\:{sin}\:{x}\:=\:\frac{\mathrm{1}}{{x}}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \:{x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \:{x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$$\int\:\frac{{sin}\:{x}}{{x}}\:{dx}\:=\:\int\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \:{x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)!}\:+\:{c} \\ $$

Commented by Tawakalitu. last updated on 27/Aug/16

Thank you for your time. i really appreciate

$${Thank}\:{you}\:{for}\:{your}\:{time}.\:{i}\:{really}\:{appreciate} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com