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Question Number 102911 by I want to learn more last updated on 11/Jul/20

∫ ((sin(x))/x)  dx

$$\int\:\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{x}}\:\:\mathrm{dx} \\ $$

Commented by prakash jain last updated on 11/Jul/20

This question has been asked so many  times. Among most recent is Q98713

$$\mathrm{This}\:\mathrm{question}\:\mathrm{has}\:\mathrm{been}\:\mathrm{asked}\:\mathrm{so}\:\mathrm{many} \\ $$$$\mathrm{times}.\:\mathrm{Among}\:\mathrm{most}\:\mathrm{recent}\:\mathrm{is}\:\mathrm{Q98713} \\ $$

Answered by Dwaipayan Shikari last updated on 11/Jul/20

∫(1/x)(x−(x^3 /(3!))+(x^5 /(5!))−(x^7 /(7!))+....)dx  ∫1−(x^2 /(3!))+(x^4 /(5!))−(x^6 /(7!))+.......dx  x−(x^3 /(3.3!))+(x^5 /(5.5!))−(x^7 /(7.7!))+...=Σ_(n=1) ^∞ (−1)^(n+1) (x^(2n−1) /((2n−1)(2n−1)!))

$$\int\frac{\mathrm{1}}{{x}}\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+....\right){dx} \\ $$$$\int\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{6}} }{\mathrm{7}!}+.......{dx} \\ $$$${x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}.\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}.\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}.\mathrm{7}!}+...=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \frac{{x}^{\mathrm{2}{n}−\mathrm{1}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)!} \\ $$

Commented by I want to learn more last updated on 11/Jul/20

Thanks sir

$$\mathrm{Thanks}\:\mathrm{sir} \\ $$

Answered by Aziztisffola last updated on 11/Jul/20

Commented by I want to learn more last updated on 12/Jul/20

Thanks sir

$$\mathrm{Thanks}\:\mathrm{sir} \\ $$

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