sin-x-x-dx-

Question Number 98713 by M±th+et+s last updated on 15/Jun/20

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

Commented by M±th+et+s last updated on 15/Jun/20

$${i}\:{have}\:{a}\:{solution}\:{i}\:{will}\:{post}\:{it}\:{later}\: \\ $$$${with}\:{using} \\ $$$${generalized}\:{hypergeometric}\:{function} \\ $$$${but}\:{i}\:{want}\:{to}\:{see}\:{if}\:\:{there}\:{were}\:{any} \\ $$$${ideas}\:{for}\:{this}\:{important}\:{integral} \\ $$

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.

$$\boldsymbol{{okk}}\:\boldsymbol{{then}},\:\boldsymbol{{I}}\:\boldsymbol{{have}}\:\boldsymbol{{another}}\:\boldsymbol{{way}} \\ $$$$\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)!}\int\boldsymbol{{x}}^{\mathrm{2}\boldsymbol{{n}}} \boldsymbol{{dx}} \\ $$$$=\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)!}.\frac{\boldsymbol{{x}}^{\mathrm{2}\boldsymbol{{n}}+\mathrm{1}} }{\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)}\:\:\:+\boldsymbol{{C}} \\ $$$$\boldsymbol{{but}}\:\boldsymbol{{it}}\:\boldsymbol{{will}}\:\boldsymbol{{be}}\:\boldsymbol{{more}}\:\boldsymbol{{beautiful}}\:\boldsymbol{{if}} \\ $$$$\boldsymbol{{you}}\:\boldsymbol{{integrate}}\:\boldsymbol{{this}}\:\boldsymbol{{within}}−\infty \\ $$$${to}+\infty. \\ $$$$\:\:\boldsymbol{{like}}\:\:\frac{\mathrm{1}}{\boldsymbol{\pi}}\int_{−\infty} ^{+\infty} \frac{\boldsymbol{{sinx}}}{\boldsymbol{{x}}}\boldsymbol{{dx}}=\mathrm{1}\:\:\:\:\:\:\:,\boldsymbol{{then}} \\ $$$$\boldsymbol{{it}}\:\boldsymbol{{behave}}\:\boldsymbol{{likes}}\:\boldsymbol{{D}}{i}\boldsymbol{{rac}}\:\boldsymbol{{delta}}\:\boldsymbol{{f}}^{\boldsymbol{{n}}} \\ $$$$\int_{−\infty} ^{+\infty} \boldsymbol{\delta}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}=\mathrm{1}\left[\boldsymbol{\delta}_{\boldsymbol{{n}}} \left(\boldsymbol{{x}}\right)=\frac{\boldsymbol{{sin}}\left(\boldsymbol{{nx}}\right)}{\boldsymbol{\pi{x}}}\right] \\ $$$$\boldsymbol{{this}}\:\boldsymbol{{beautiful}}\:\boldsymbol{{result}}\:\:\boldsymbol{{is}}\:\boldsymbol{{very}} \\ $$$$\boldsymbol{{useful}}\:\boldsymbol{{in}}\:\boldsymbol{{Quantum}}\:\boldsymbol{{Mechanics}}. \\ $$

Commented by M±th+et+s last updated on 15/Jun/20

$${very}\:{nice}\:{sir}\:{thank}\:{you}\: \\ $$

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

$${I}=\int\frac{\mathrm{1}}{{x}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}{dx}=\int\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\gamma−\gamma} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}{dx} \\ $$$${I}=\int\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)!}+{c} \\ $$$${I}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}\right)!}+{c}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2}{n}\right)!} \\ $$$$\left(\mathrm{2}{n}\right)!=\mathrm{2}^{{n}} .{n}!\left(\mathrm{2}{n}−\mathrm{1}\right)!! \\ $$$${I}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right)^{\mathrm{2}} \frac{\left(−\mathrm{1}\right)^{{n}} \ast\left({x}\right)\ast\left({x}^{\mathrm{2}{n}} \right)}{\mathrm{2}^{{n}} {n}!\left(\mathrm{2}{n}−\mathrm{1}\right)!!}+{c}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right)^{\mathrm{2}} \frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{{n}} {n}!\left(\mathrm{2}{n}−\mathrm{1}\right)!!}+{c} \\ $$$${I}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\left(\mathrm{2}{n}\right)!}{\left(\mathrm{2}{n}\right)!\left(\mathrm{2}{n}+\mathrm{1}\right)}\right)^{\mathrm{2}} \frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{{n}} \:{n}!\ast\frac{\mathrm{2}^{{n}} }{\mathrm{2}^{{n}} }\left(\mathrm{2}{n}−\mathrm{1}\right)!!}+{c} \\ $$$${I}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}^{{n}} \mu!\left(\mathrm{2}{n}−\mathrm{1}\right)!!}{\mathrm{2}^{{n}} \mu!\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)!!}\right)^{\mathrm{2}} \frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{{n}} {n}!\ast\frac{\mathrm{2}^{{n}} }{\mathrm{2}^{{n}} }\left(\mathrm{2}{n}−\mathrm{1}\right)!!}+{c} \\ $$$$\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)!!=\left(\mathrm{2}{n}+\mathrm{1}\right)!! \\ $$$${I}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)!!}{\mathrm{2}^{{n}} }\:}{\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)!!}{\mathrm{2}^{{n}} }}\right]^{\mathrm{2}} \frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{4}^{{n}} \:{n}!\ast\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)!!}{\mathrm{2}^{{n}} }}+{c} \\ $$$${I}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:_{{n}} }{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{{n}} }\right]^{\mathrm{2}} \frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{n}} }\:\frac{\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{4}}\right)^{{n}} }{{n}!}+{c}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:_{{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:_{{n}} }{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{{n}} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{{n}} }\:\frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:_{{n}} }\:\frac{\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{4}}\right)^{{n}} }{{n}!}+{c} \\ $$$${I}={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:_{{n}} }{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{{n}} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{{n}} }\:\frac{\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{4}}\right)^{{n}} \:}{{n}!}+{c}={x}\:\mathrm{1}{F}\mathrm{2}\left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\frac{−{x}^{\mathrm{2}} }{\mathrm{4}}\right]+{c} \\ $$$$ \\ $$$${finaly}\:\int\frac{{sin}\left({x}\right)}{{x}}{dx}={x}\:\mathrm{1}{F}\mathrm{2}\left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\frac{−{x}^{\mathrm{2}} }{\mathrm{4}}\right]+{c}\: \\ $$

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))

$$\left({a}\right)\:_{{n}} :\:{is}\:{pochhamer}\:{sympol} \\ $$$$\mathrm{1}{F}\mathrm{2}:\:{is}\:{special}\:{function}\:{called}\: \\ $$$$\left(\left({generalized}\:{hyper}\:{geometric}\:{function}\right)\right) \\ $$$$ \\ $$

Commented by smridha last updated on 16/Jun/20