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Question Number 102701 by 175mohamed last updated on 10/Jul/20
Evaluate: ∫((sin x)/x)dx
Evaluate:∫sinxxdx
Commented by  M±th+et+s last updated on 10/Jul/20
go to Q.98713 i solved this question
gotoQ.98713isolvedthisquestion
Answered by PRITHWISH SEN 2 last updated on 10/Jul/20
∫(1/x){x−(x^3 /(3!)) +(x^5 /(5!))−...}dx = Σ_(r=1) ^∞ (−1)^(r+1) (x^(2r−1) /((2r−1).(2r−1)!)) please check.
∫1x{x−x33!+x55!−…}dx=∑∞r=1(−1)r+1x2r−1(2r−1).(2r−1)!pleasecheck.
Answered by mathmax by abdo last updated on 10/Jul/20
at form of serie wehave sinx =Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)!)) with radius r=+∞ ⇒((sinx)/x) =Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2n+1)!)) ⇒ ∫ ((sinx)/x)dx =Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!)) ∫ x^(2n) dx =Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!(2n+1)))x^(2n+1) +C
atformofseriewehavesinx=∑n=0∞(−1)nx2n+1(2n+1)!withradiusr=+∞⇒sinxx=∑n=0∞(−1)nx2n(2n+1)!⇒∫sinxxdx=∑n=0∞(−1)n(2n+1)!∫x2ndx=∑n=0∞(−1)n(2n+1)!(2n+1)x2n+1+C

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