Question Number 46847 by maxmathsup by imad last updated on 01/Nov/18
$${calculate}\:\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}} \:\:{e}^{\frac{{x}}{{y}}} {dxdy} \\ $$
Commented by maxmathsup by imad last updated on 04/Nov/18
![I =∫_1 ^2 (∫_0 ^1 e^(x/y) dx)dy but ∫_0 ^1 e^(x/y) dx =[y e^(x/y) ]_(x=0) ^(x =1) = y(e^(1/y) −1) ⇒ I =∫_1 ^2 (y e^(1/y) )dy −∫_1 ^2 ydy but ∫_1 ^2 ydy =[(y^2 /2)]_1 ^2 =2−(1/2) =(3/2) and ∫_1 ^2 y e^(1/y) dy =_((1/y)=t) ∫_(1/2) ^1 (1/t) e^t (dt/t^2 ) =∫_(1/2) ^1 (e^t /t^3 )dt =∫_(1/2) ^1 t^(−3) e^t dt =[−(1/2) t^(−2) e^t ]_(1/2) ^1 +∫_(1/2) ^1 (1/2) t^(−2) e^t dt =(1/2){ 4 e^(1/2) −e} +(1/2){ [−(1/t) e^t ]_(1/2) ^1 +∫_(1/2) ^1 (1/t) e^t dt} 2(√e)−(e/2) +(1/2){2(√e)−e + ∫_(1/2) ^1 (e^t /t) dt} =3(√e)−e +(1/2) ∫_(1/2) ^1 (e^t /t) dt and ∫_(1/2) ^1 (e^t /t) dt =∫_(1/2) ^1 (1/t)(Σ_(n=0) ^∞ (t^n /(n!)))dt =∫_(1/2) ^1 (1/t)(1+Σ_(n=1) ^∞ (t^n /(n!)))dt =ln(2) +Σ_(n=1) ^∞ (1/(n!)) ∫_(1/2) ^1 t^(n−1) dt =ln(2)+Σ_(n=1) ^∞ (1/(n.(n!)))(1−(1/2^n )) ...be continued...](https://www.tinkutara.com/question/Q47117.png)
$${I}\:=\int_{\mathrm{1}} ^{\mathrm{2}} \:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{\frac{{x}}{{y}}} {dx}\right){dy}\:\:{but}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{\frac{{x}}{{y}}} \:{dx}\:=\left[{y}\:{e}^{\frac{{x}}{{y}}} \right]_{{x}=\mathrm{0}} ^{{x}\:=\mathrm{1}} =\:{y}\left({e}^{\frac{\mathrm{1}}{{y}}} −\mathrm{1}\right)\:\:\Rightarrow \\ $$$${I}\:=\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({y}\:{e}^{\frac{\mathrm{1}}{{y}}} \right){dy}\:−\int_{\mathrm{1}} ^{\mathrm{2}} {ydy}\:\:{but}\:\int_{\mathrm{1}} ^{\mathrm{2}} {ydy}\:=\left[\frac{{y}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{1}} ^{\mathrm{2}} \:=\mathrm{2}−\frac{\mathrm{1}}{\mathrm{2}}\:=\frac{\mathrm{3}}{\mathrm{2}}\:{and} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \:{y}\:{e}^{\frac{\mathrm{1}}{{y}}} {dy}\:=_{\frac{\mathrm{1}}{{y}}={t}} \:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{t}}\:{e}^{{t}} \:\frac{{dt}}{{t}^{\mathrm{2}} }\:=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{e}^{{t}} }{{t}^{\mathrm{3}} }{dt}\:\:=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:{t}^{−\mathrm{3}} \:{e}^{{t}} \:{dt}\: \\ $$$$=\left[−\frac{\mathrm{1}}{\mathrm{2}}\:{t}^{−\mathrm{2}} \:{e}^{{t}} \right]_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:+\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{2}}\:{t}^{−\mathrm{2}} \:{e}^{{t}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\mathrm{4}\:{e}^{\frac{\mathrm{1}}{\mathrm{2}}} −{e}\right\}\:\:+\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\left[−\frac{\mathrm{1}}{{t}}\:{e}^{{t}} \right]_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:+\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}}{{t}}\:{e}^{{t}} {dt}\right\} \\ $$$$\mathrm{2}\sqrt{{e}}−\frac{{e}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}\left\{\mathrm{2}\sqrt{{e}}−{e}\:+\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{e}^{{t}} }{{t}}\:{dt}\right\} \\ $$$$=\mathrm{3}\sqrt{{e}}−{e}\:\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{e}^{{t}} }{{t}}\:{dt}\:\:{and}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{e}^{{t}} }{{t}}\:{dt}\:=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{t}}\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{t}^{{n}} }{{n}!}\right){dt} \\ $$$$=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{t}}\left(\mathrm{1}+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{t}^{{n}} }{{n}!}\right){dt}\:={ln}\left(\mathrm{2}\right)\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:{t}^{{n}−\mathrm{1}} {dt} \\ $$$$={ln}\left(\mathrm{2}\right)+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}.\left({n}!\right)}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\right)\:…{be}\:{continued}… \\ $$