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Question Number 98713 by  M±th+et+s last updated on 15/Jun/20
∫((sin(x))/x)dx
∫sin(x)xdx
Commented by  M±th+et+s last updated on 15/Jun/20
i have a solution i will post it later with using generalized hypergeometric function but i want to see if there were any ideas for this important integral
ihaveasolutioniwillpostitlaterwithusinggeneralizedhypergeometricfunctionbutiwanttoseeiftherewereanyideasforthisimportantintegral
Answered by smridha last updated on 15/Jun/20
okk then, I have another way Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))∫x^(2n) dx =Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!)).(x^(2n+1) /((2n+1))) +C but it will be more beautiful if you integrate this within−∞ to+∞. like (1/𝛑)∫_(−∞) ^(+∞) ((sinx)/x)dx=1 ,then it behave likes Dirac delta f^n ∫_(−∞) ^(+∞) 𝛅(x)dx=1[𝛅_n (x)=((sin(nx))/(𝛑x))] this beautiful result is very useful in Quantum Mechanics.
okkthen,Ihaveanotherway∑∞n=0(−1)n(2n+1)!∫x2ndx=∑∞n=0(−1)n(2n+1)!.x2n+1(2n+1)+Cbutitwillbemorebeautifulifyouintegratethiswithin−∞to+∞.like1π∫−∞+∞sinxxdx=1,thenitbehavelikesDiracdeltafn∫−∞+∞δ(x)dx=1[δn(x)=sin(nx)πx]thisbeautifulresultisveryusefulinQuantumMechanics.
Commented by  M±th+et+s last updated on 15/Jun/20
very nice sir thank you
verynicesirthankyou
Answered by  M±th+et+s last updated on 16/Jun/20
I=∫(1/x)Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)!))dx=∫Σ_(n=0) ^∞ (((−1)^n x^(2n+γ−γ) )/((2n+1)!))dx I=∫Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2n+1)!))dx=Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)(2n+1)!))+c I=Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)(2n+1)(2n)!))+c=Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/((2n+1)^2 (2n)!)) (2n)!=2^n .n!(2n−1)!! I=Σ_(n=0) ^∞ ((1/(2n+1)))^2 (((−1)^n ∗(x)∗(x^(2n) ))/(2^n n!(2n−1)!!))+c=xΣ_(n=0) ^∞ ((1/(2n+1)))^2 (((−1)^n x^(2n) )/(2^n n!(2n−1)!!))+c I=xΣ_(n=0) ^∞ ((((2n)!)/((2n)!(2n+1))))^2 (((−1)^n x^(2n) )/(2^n n!∗(2^n /2^n )(2n−1)!!))+c I=xΣ_(n=0) ^∞ (((2^n μ!(2n−1)!!)/(2^n μ!(2n+1)(2n−1)!!)))^2 (((−1)^n x^(2n) )/(2^n n!∗(2^n /2^n )(2n−1)!!))+c (2n+1)(2n−1)!!=(2n+1)!! I=xΣ_(n=0) ^∞ [(((((2n−1)!!)/2^n ) )/(((2n+1)!!)/2^n ))]^2 (((−1)^n x^(2n) )/(4^n n!∗(((2n−1)!!)/2^n )))+c I=xΣ_(n=0) ^∞ [((((1/2)) _n )/(((3/2)) _n ))]^2 (1/(((1/2))^n )) (((((−x^2 )/4))^n )/(n!))+c=xΣ_(n=0) ^∞ ((((1/2)) _n ((1/2)) _n )/(((3/2)) _n ((3/2)) _n )) (1/(((1/2)) _n )) (((((−x^2 )/4))^n )/(n!))+c I=xΣ_(n=0) ^∞ ((((1/2)) _n )/(((3/2)) _n ((3/2)) _n )) (((((−x^2 )/4))^n )/(n!))+c=x 1F2[(1/2),(3/2);(3/2);((−x^2 )/4)]+c finaly ∫((sin(x))/x)dx=x 1F2[(1/2),(3/2);(3/2);((−x^2 )/4)]+c
I=∫1x∑∞n=0(−1)nx2n+1(2n+1)!dx=∫∑∞n=0(−1)nx2n+γ−γ(2n+1)!dxI=∫∑∞n=0(−1)nx2n(2n+1)!dx=∑∞n=0(−1)nx2n+1(2n+1)(2n+1)!+cI=∑∞n=0(−1)nx2n+1(2n+1)(2n+1)(2n)!+c=∑∞n=0(−1)nx2n+1(2n+1)2(2n)!(2n)!=2n.n!(2n−1)!!I=∑∞n=0(12n+1)2(−1)n∗(x)∗(x2n)2nn!(2n−1)!!+c=x∑∞n=0(12n+1)2(−1)nx2n2nn!(2n−1)!!+cI=x∑∞n=0((2n)!(2n)!(2n+1))2(−1)nx2n2nn!∗2n2n(2n−1)!!+cI=x∑∞n=0(2nμ!(2n−1)!!2nμ!(2n+1)(2n−1)!!)2(−1)nx2n2nn!∗2n2n(2n−1)!!+c(2n+1)(2n−1)!!=(2n+1)!!I=x∑∞n=0[(2n−1)!!2n(2n+1)!!2n]2(−1)nx2n4nn!∗(2n−1)!!2n+cI=x∑∞n=0[(12)n(32)n]21(12)n(−x24)nn!+c=x∑∞n=0(12)n(12)n(32)n(32)n1(12)n(−x24)nn!+cI=x∑∞n=0(12)n(32)n(32)n(−x24)nn!+c=x1F2[12,32;32;−x24]+cfinaly∫sin(x)xdx=x1F2[12,32;32;−x24]+c
Commented by  M±th+et+s last updated on 16/Jun/20
(a) _n : is pochhamer sympol 1F2: is special function called ((generalized hyper geometric function))
(a)n:ispochhamersympol1F2:isspecialfunctioncalled((generalizedhypergeometricfunction))
Commented by smridha last updated on 16/Jun/20
great manipulation..
greatmanipulation..
Commented by  M±th+et+s last updated on 16/Jun/20
thank you sir
thankyousir
Commented by smridha last updated on 16/Jun/20
welcome..
welcome..

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