Question Number 39975 by sanjar last updated on 14/Jul/18
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{{x}} \:{dx}\right)^{\mathrm{2}} }{\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } \:{dx}}\:\:= \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jul/18
$${is}\:\int_{\mathrm{0}} ^{{x}} {e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:{intregable}… \\ $$
Commented by maxmathsup by imad last updated on 14/Jul/18
$${for}\:{x}\:{fixed}\:\:{the}\:{integral}\:{exist}\:{but}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:{is}\:{divergent}… \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jul/18
$${yes}\:{here}\:{x}\rightarrow\infty\:{so}\:{it}\:{is}\:{diverging}…{you}\:{are}\:{right} \\ $$$${sir}… \\ $$
Commented by abdo mathsup 649 cc last updated on 14/Jul/18
![if g and f are intehrable on X we hsve ∫_X fgdx≤ (∫_X f^2 dx)^(1/2) .(∫_X g^2 dx)^(1/2) (iequqlity of cauchy shwarz) ⇒ (∫_X fgdx)^2 ≤ (∫_X f^2 dx).(∫_X g^2 dx) let take f=e^x and g=1 ⇒(∫_0 ^x e^x dx)^2 ≤ x ∫_0 ^x e^(2x) dx also (∫_0 ^x e^x^2 dx)^2 ≤ x^2 ∫_0 ^x e^(2x^2 ) dx⇒ ∫_0 ^x e^(2x^2 ) dx ≥ (1/x^2 ) (∫_0 ^x e^x^2 dx)^2 ⇒ (1/(∫_0 ^x e^(2x^2 ) dx)) ≤ (x^2 /(( ∫_0 ^x e^x^2 dx)^2 )) ⇒ (((∫_0 ^x e^x dx)^(2i) )/(∫_0 ^x e^(2x^2 ) dx)) ≤ x^3 ((∫_0 ^x e^(2x) dx)/(( ∫_0 ^x e^x^2 dx)^2 )) →?(x→+∞) let S_n = ∫_0 ^n e^(2x) dx =Σ_(k=0) ^(n−1) ∫_k ^(k+1) e^(2x) dx =(1/2)Σ_(k=0) ^(n−1) {e^(2(k+1)) −e^(2k) } W_n = ∫_0 ^n e^x^2 dx =Σ_(k=0) ^(n−1) ∫_k ^(k+1) e^x^2 dx ∫_k ^(k+1) e^x^2 dx =(1/2) ∫_k ^(k+1) 2x e^x^2 (dx/x) =(1/2){ [(e^x^2 /x)]_k ^(k+1) − ∫_k ^(k+1) e^x^2 (−(1/x^2 ))dx} =(1/2){ (e^((k+1)^2 ) /(k+1)) − (e^k^2 /k) + ∫_k ^(k+1) (e^x^2 /x^2 )dx}....be continued...](https://www.tinkutara.com/question/Q39998.png)
$${if}\:{g}\:{and}\:{f}\:{are}\:{intehrable}\:{on}\:{X}\:\:{we}\:{hsve} \\ $$$$\int_{{X}} {fgdx}\leqslant\:\left(\int_{{X}} {f}^{\mathrm{2}} {dx}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:.\left(\int_{{X}} {g}^{\mathrm{2}} {dx}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left({iequqlity}\:{of}\right. \\ $$$$\left.{cauchy}\:{shwarz}\right)\:\Rightarrow \\ $$$$\left(\int_{{X}} {fgdx}\right)^{\mathrm{2}} \leqslant\:\left(\int_{{X}} {f}^{\mathrm{2}} {dx}\right).\left(\int_{{X}} {g}^{\mathrm{2}} {dx}\right)\:{let}\:{take}\:{f}={e}^{{x}} \\ $$$${and}\:{g}=\mathrm{1}\:\Rightarrow\left(\int_{\mathrm{0}} ^{{x}} {e}^{{x}} {dx}\right)^{\mathrm{2}} \leqslant\:{x}\:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}} {dx}\:{also} \\ $$$$\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} \:\leqslant\:{x}^{\mathrm{2}} \:\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:\geqslant\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} \:\Rightarrow\:\frac{\mathrm{1}}{\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}}\:\leqslant\:\frac{{x}^{\mathrm{2}} }{\left(\:\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }\:\Rightarrow \\ $$$$\frac{\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}} {dx}\right)^{\mathrm{2}{i}} }{\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}}\:\leqslant\:\:{x}^{\mathrm{3}} \:\:\:\frac{\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}} {dx}}{\left(\:\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }\:\rightarrow?\left({x}\rightarrow+\infty\right) \\ $$$${let}\:{S}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:{e}^{\mathrm{2}{x}} {dx}\:\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{\mathrm{2}{x}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left\{{e}^{\mathrm{2}\left({k}+\mathrm{1}\right)} \:−{e}^{\mathrm{2}{k}} \right\} \\ $$$${W}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{e}^{{x}^{\mathrm{2}} } {dx}\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } {dx} \\ $$$$\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } {dx}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\mathrm{2}{x}\:{e}^{{x}^{\mathrm{2}} } \:\frac{{dx}}{{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\left[\frac{{e}^{{x}^{\mathrm{2}} } }{{x}}\right]_{{k}} ^{{k}+\mathrm{1}} \:−\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } \left(−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){dx}\right\}\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\frac{{e}^{\left({k}+\mathrm{1}\right)^{\mathrm{2}} } }{{k}+\mathrm{1}}\:−\:\frac{{e}^{{k}^{\mathrm{2}} } }{{k}}\:\:+\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\frac{{e}^{{x}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx}\right\}….{be}\:{continued}… \\ $$