Question Number 6132 by sanusihammed last updated on 15/Jun/16
![Evaluate the integral of ... [(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 < ∞ The answer is saying ............ (√e) How is the answer (√e)](https://www.tinkutara.com/question/Q6132.png)
$${Evaluate}\:{the}\:{integral}\:{of}\:… \\ $$$$ \\ $$$$\left[\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{{x}^{\mathrm{5}} }{\mathrm{2}.\mathrm{4}}−\frac{{x}^{\mathrm{7}} }{\mathrm{2}.\mathrm{4}.\mathrm{6}}+….\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} }−\frac{{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} .\mathrm{6}^{\mathrm{2}} }+….\right)\right]{dx} \\ $$$$ \\ $$$${for}\:\:\mathrm{0}\:<\:\:{x}\:\:<\:\:\infty \\ $$$$ \\ $$$${The}\:{answer}\:{is}\:{saying}\:\:…………\:\:\sqrt{{e}} \\ $$$$ \\ $$$${How}\:{is}\:{the}\:{answer}\:\:\sqrt{{e}} \\ $$
Commented by prakash jain last updated on 15/Jun/16
$${x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{{x}^{\mathrm{5}} }{\mathrm{2}\centerdot\mathrm{4}}={x}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{4}} }{\mathrm{2}\centerdot\mathrm{4}}−+…\right) \\ $$$$={x}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} /\mathrm{2}}{\mathrm{1}!}+\frac{\left({x}^{\mathrm{2}} /\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{2}!}−\frac{\left({x}^{\mathrm{2}} /\mathrm{2}\right)^{\mathrm{3}} }{\mathrm{3}!}\right) \\ $$$$={xe}^{−{x}^{\mathrm{2}} /\mathrm{2}} \\ $$$$\mathrm{similar}\:\mathrm{you}\:\mathrm{can}\:\mathrm{try}\:\mathrm{evaluating}\:\mathrm{other}\:\mathrm{expression} \\ $$$$\mathrm{and}\:\mathrm{integrate}. \\ $$