Question Number 5835 by sanusihammed last updated on 31/May/16
![Evaluate the integral. ∫[(x−(x^3 /2)+(x^5 /(2.4))−(x^7 /(2.4.6))+...)(1−(x^2 /2^2 )+(x^4 /(2^2 .4^2 ))−(x^6 /(2^2 .4^2 .6^2 ))+....)]dx for 0<x<∞ Please help](https://www.tinkutara.com/question/Q5835.png)
$${Evaluate}\:{the}\:{integral}. \\ $$$$ \\ $$$$\int\left[\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{{x}^{\mathrm{5}} }{\mathrm{2}.\mathrm{4}}−\frac{{x}^{\mathrm{7}} }{\mathrm{2}.\mathrm{4}.\mathrm{6}}+…\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} }−\frac{{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} .\mathrm{6}^{\mathrm{2}} }+….\right)\right]{dx} \\ $$$$ \\ $$$${for}\:\:\:\:\mathrm{0}<{x}<\infty \\ $$$$ \\ $$$${Please}\:{help} \\ $$
Commented by Yozzii last updated on 31/May/16
![(1/((2n−2)!!))=(1/(2×4×6×8×10×...×(2n−2))) =(1/(2^(n−1) (1×2×3×4×5×...×(n−1)))) ∴(1/((2n)!!))=(1/(2^n n!)) ( n≥0) (2n)!!=2n{(2n−2)!!} Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(2^n (n!)))=(√2)Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(((√2))^(2n+1) n!)) =xΣ_(n=0) ^∞ (((−1)^n )/(n!))((x^2 /2))^n =xΣ_(n=0) ^∞ (1/(n!))(−(x^2 /2))^n Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(2^n (n!)))=xexp(((−x^2 )/2))=−(d/dx)(e^(−x^2 /2) ) −−−−−−−−−−−−−−−−−−−−−−− 1−(x^2 /2^2 )+(x^4 /(2^2 4^2 ))−(x^6 /(2^2 4^2 6^2 ))+...=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2^n n!)^2 )) =Σ_(n=0) ^∞ (1/((n!)^2 ))(−(x^2 /4))^n =Σ_(n=0) ^∞ {(1/(n!))(((−x)/2))^n }{(1/(n!))((x/2))^n } −−−−−−−−−−−−−−−−−−−−−− ∴J=∫{xexp(−0.5x^2 )(Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2^n n!)^2 )))}dx J=Σ_(n=0) ^∞ [(((−1)^n )/((2^n n!)^2 )){∫(xe^(−0.5x^2 ) )x^(2n) dx}] =Σ_(n=0) ^∞ (((−1)^n )/((2^n n!)^2 )){−e^(−0.5x^2 ) x^(2n) +2n∫xe^(−0.5x^2 ) x^(2n−2) dx} =Σ_(n=0) ^∞ (((−1)^n )/((2^n n!)^2 ))(−x^(2n) e^(−0.5x^2 ) −2nx^(2n−2) e^(−0.5x^2 ) −2n(2n−2)x^(2n−4) e^(−0.5x^2 ) +2n(2n−2)(2n−4)∫x^(2n−6) xe^(−0.5x^2 ) dx) =Σ_(n=0) ^∞ (((−1)^n )/(((2n)!!)^2 ))(−e^(−0.5x^2 ) (x^(2n) +2nx^(2(n−1)) +2n(2n−2)x^(2(n−2)) +2n(2n−2)(2n−4)x^(2(n−3)) +....+(2n)!!x^(2(n−n)) ))+C J=Σ_(n=0) ^∞ ((((−1)^(n+1) e^(−0.5x^2 ) )/(((2n)!!)^2 )){Σ_(k=0) ^n x^(2(n−k)) (((2n)!!)/((2(n−k))!!))})+C J=e^(−0.5x^2 ) Σ_(n=0) ^∞ ((((−1)^(n+1) )/((2n)!!)){Σ_(k=0) ^n (1/((2(n−k))!!))x^(2(n−k)) })+C J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ ((1/(n!))(((−x^2 )/2))^n {Σ_(k=0) ^n (1/((n−k)!))((x^2 /2))^(n−k) }) J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ Σ_(k=0) ^n {(1/(n!(n−k)!))(((−x^2 )/2))^n ((x^2 /2))^(n−k) } J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ ((1/(n!))(((−x^2 )/2))^n )(Σ_(k=0) ^∞ (1/((n−k)!))((x^2 /2))^(n−k) )+C ??? J=−e^(−0.5x^2 ) e^(−0.5x^2 ) e^(0.5x^2 ) +C J=−e^(−0.5x^2 ) +C J∣_0 ^∞ =−e^(−0.5x^2 ) ∣_0 ^∞ =1.](https://www.tinkutara.com/question/Q5837.png)
$$\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{2}\right)!!}=\frac{\mathrm{1}}{\mathrm{2}×\mathrm{4}×\mathrm{6}×\mathrm{8}×\mathrm{10}×…×\left(\mathrm{2}{n}−\mathrm{2}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{{n}−\mathrm{1}} \left(\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{4}×\mathrm{5}×…×\left({n}−\mathrm{1}\right)\right)} \\ $$$$\therefore\frac{\mathrm{1}}{\left(\mathrm{2}{n}\right)!!}=\frac{\mathrm{1}}{\mathrm{2}^{{n}} {n}!}\:\:\:\:\left(\:{n}\geqslant\mathrm{0}\right) \\ $$$$\left(\mathrm{2}{n}\right)!!=\mathrm{2}{n}\left\{\left(\mathrm{2}{n}−\mathrm{2}\right)!!\right\} \\ $$$$ \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{2}^{{n}} \left({n}!\right)}=\sqrt{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}{n}+\mathrm{1}} {n}!} \\ $$$$={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}} \\ $$$$={x}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\left(−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{2}^{{n}} \left({n}!\right)}={xexp}\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{2}}\right)=−\frac{{d}}{{dx}}\left({e}^{−{x}^{\mathrm{2}} /\mathrm{2}} \right) \\ $$$$−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} \mathrm{4}^{\mathrm{2}} }−\frac{{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} \mathrm{4}^{\mathrm{2}} \mathrm{6}^{\mathrm{2}} }+…=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\left(\mathrm{2}^{{n}} {n}!\right)^{\mathrm{2}} } \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}!\right)^{\mathrm{2}} }\left(−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)^{{n}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{{n}!}\left(\frac{−{x}}{\mathrm{2}}\right)^{{n}} \right\}\left\{\frac{\mathrm{1}}{{n}!}\left(\frac{{x}}{\mathrm{2}}\right)^{{n}} \right\} \\ $$$$−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\therefore{J}=\int\left\{{xexp}\left(−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} \right)\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\left(\mathrm{2}^{{n}} {n}!\right)^{\mathrm{2}} }\right)\right\}{dx} \\ $$$${J}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}^{{n}} {n}!\right)^{\mathrm{2}} }\left\{\int\left({xe}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \right){x}^{\mathrm{2}{n}} {dx}\right\}\right] \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}^{{n}} {n}!\right)^{\mathrm{2}} }\left\{−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } {x}^{\mathrm{2}{n}} +\mathrm{2}{n}\int{xe}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } {x}^{\mathrm{2}{n}−\mathrm{2}} {dx}\right\} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}^{{n}} {n}!\right)^{\mathrm{2}} }\left(−{x}^{\mathrm{2}{n}} {e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } −\mathrm{2}{nx}^{\mathrm{2}{n}−\mathrm{2}} {e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } −\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{2}\right){x}^{\mathrm{2}{n}−\mathrm{4}} {e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } +\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{4}\right)\int{x}^{\mathrm{2}{n}−\mathrm{6}} {xe}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } {dx}\right) \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\left(\mathrm{2}{n}\right)!!\right)^{\mathrm{2}} }\left(−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \left({x}^{\mathrm{2}{n}} +\mathrm{2}{nx}^{\mathrm{2}\left({n}−\mathrm{1}\right)} +\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{2}\right){x}^{\mathrm{2}\left({n}−\mathrm{2}\right)} +\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{4}\right){x}^{\mathrm{2}\left({n}−\mathrm{3}\right)} +….+\left(\mathrm{2}{n}\right)!!{x}^{\mathrm{2}\left({n}−{n}\right)} \right)\right)+{C} \\ $$$${J}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } }{\left(\left(\mathrm{2}{n}\right)!!\right)^{\mathrm{2}} }\left\{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{x}^{\mathrm{2}\left({n}−{k}\right)} \frac{\left(\mathrm{2}{n}\right)!!}{\left(\mathrm{2}\left({n}−{k}\right)\right)!!}\right\}\right)+{C} \\ $$$${J}={e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\left(\mathrm{2}{n}\right)!!}\left\{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}\left({n}−{k}\right)\right)!!}{x}^{\mathrm{2}\left({n}−{k}\right)} \right\}\right)+{C} \\ $$$${J}=−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}!}\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}} \left\{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\left({n}−{k}\right)!}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}−{k}} \right\}\right) \\ $$$${J}=−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left\{\frac{\mathrm{1}}{{n}!\left({n}−{k}\right)!}\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}−{k}} \right\} \\ $$$$ \\ $$$${J}=−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}!}\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}} \right)\left(\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}−{k}\right)!}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)^{{n}−{k}} \right)+{C}\:\:??? \\ $$$${J}=−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } {e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } {e}^{\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } +{C} \\ $$$${J}=−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } +{C} \\ $$$${J}\mid_{\mathrm{0}} ^{\infty} =−{e}^{−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} } \mid_{\mathrm{0}} ^{\infty} =\mathrm{1}. \\ $$
Commented by sanusihammed last updated on 31/May/16
$${Thanks}\:{so}\:{much} \\ $$