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Question Number 39975 by sanjar last updated on 14/Jul/18

lim_(x→∞)  (((∫_( 0) ^x   e^x  dx)^2 )/(∫_( 0) ^x   e^(2x^2 )  dx))  =

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{{x}} \:{dx}\right)^{\mathrm{2}} }{\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } \:{dx}}\:\:= \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jul/18

is ∫_0 ^x e^(2x^2 ) dx intregable...

$${is}\:\int_{\mathrm{0}} ^{{x}} {e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:{intregable}... \\ $$

Commented by maxmathsup by imad last updated on 14/Jul/18

for x fixed  the integral exist but  ∫_0 ^∞   e^(2x^2 ) dx is divergent...

$${for}\:{x}\:{fixed}\:\:{the}\:{integral}\:{exist}\:{but}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:{is}\:{divergent}... \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jul/18

yes here x→∞ so it is diverging...you are right  sir...

$${yes}\:{here}\:{x}\rightarrow\infty\:{so}\:{it}\:{is}\:{diverging}...{you}\:{are}\:{right} \\ $$$${sir}... \\ $$

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...

$${if}\:{g}\:{and}\:{f}\:{are}\:{intehrable}\:{on}\:{X}\:\:{we}\:{hsve} \\ $$$$\int_{{X}} {fgdx}\leqslant\:\left(\int_{{X}} {f}^{\mathrm{2}} {dx}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:.\left(\int_{{X}} {g}^{\mathrm{2}} {dx}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left({iequqlity}\:{of}\right. \\ $$$$\left.{cauchy}\:{shwarz}\right)\:\Rightarrow \\ $$$$\left(\int_{{X}} {fgdx}\right)^{\mathrm{2}} \leqslant\:\left(\int_{{X}} {f}^{\mathrm{2}} {dx}\right).\left(\int_{{X}} {g}^{\mathrm{2}} {dx}\right)\:{let}\:{take}\:{f}={e}^{{x}} \\ $$$${and}\:{g}=\mathrm{1}\:\Rightarrow\left(\int_{\mathrm{0}} ^{{x}} {e}^{{x}} {dx}\right)^{\mathrm{2}} \leqslant\:{x}\:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}} {dx}\:{also} \\ $$$$\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} \:\leqslant\:{x}^{\mathrm{2}} \:\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}\:\geqslant\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} \:\Rightarrow\:\frac{\mathrm{1}}{\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}}\:\leqslant\:\frac{{x}^{\mathrm{2}} }{\left(\:\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }\:\Rightarrow \\ $$$$\frac{\left(\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}} {dx}\right)^{\mathrm{2}{i}} }{\int_{\mathrm{0}} ^{{x}} \:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } {dx}}\:\leqslant\:\:{x}^{\mathrm{3}} \:\:\:\frac{\int_{\mathrm{0}} ^{{x}} \:{e}^{\mathrm{2}{x}} {dx}}{\left(\:\int_{\mathrm{0}} ^{{x}} \:{e}^{{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }\:\rightarrow?\left({x}\rightarrow+\infty\right) \\ $$$${let}\:{S}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:{e}^{\mathrm{2}{x}} {dx}\:\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{\mathrm{2}{x}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left\{{e}^{\mathrm{2}\left({k}+\mathrm{1}\right)} \:−{e}^{\mathrm{2}{k}} \right\} \\ $$$${W}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{e}^{{x}^{\mathrm{2}} } {dx}\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } {dx} \\ $$$$\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } {dx}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\mathrm{2}{x}\:{e}^{{x}^{\mathrm{2}} } \:\frac{{dx}}{{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\left[\frac{{e}^{{x}^{\mathrm{2}} } }{{x}}\right]_{{k}} ^{{k}+\mathrm{1}} \:−\:\int_{{k}} ^{{k}+\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } \left(−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){dx}\right\}\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\frac{{e}^{\left({k}+\mathrm{1}\right)^{\mathrm{2}} } }{{k}+\mathrm{1}}\:−\:\frac{{e}^{{k}^{\mathrm{2}} } }{{k}}\:\:+\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\frac{{e}^{{x}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx}\right\}....{be}\:{continued}... \\ $$

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