Question Number 220590 by SdC355 last updated on 16/May/25
$$\int_{\mathrm{0}} ^{\:\infty} \:\:{w}\mathrm{Ci}\left({w}\right){e}^{−{w}} \:\mathrm{d}{w}=?? \\ $$$$\mathrm{Ci}\left({w}\right)=−\int_{{w}} ^{\:\infty} \:\:\frac{\mathrm{cos}\left({t}\right)}{{t}}\:\mathrm{d}{t} \\ $$
Answered by breniam last updated on 17/May/25
![∫_0 ^∞ wCi(w)e^(−w) dw ∫we^(−w) dw=−we^(−w) −e^(−w) ∫cos(w)e^(−w) dw=∫(sin(w))′e^(−w) dw= sin(w)e^(−w) +∫sin(w)e^(−w) dw= sin(w)e^(−w) −∫(cos(w))′e^(−w) dw= sin(w)e^(−w) −cos(w)e^(−w) −∫cos(w)e^(−w) dw⇒ ⇒∫cos(w)e^(−w) dw=((sin(w)e^(−w) −cos(w)e^(−w) )/2) lim_(x→0^+ ) [xCi(x)]=lim_(x→0^+ ) [((Ci(x))/(1/x))]=^H lim_(x→0^+ ) [(((cos(x))/x)/(−(1/x^2 )))]=0 ∫_0 ^∞ wCi(w)e^(−w) dw=−∫_0 ^∞ (we^(−w) +e^(−w) )′Ci(w)dw= lim_(x→0^+ ) [e^(−x) Ci(x)+∫_x ^∞ cos(w)e^(−w) dw+∫_x ^∞ ((cos(w)e^(−w) )/w)dw]= lim_(x→0^+ ) [e^(−x) Ci(x)−((sin(x)e^(−x) −cos(x)e^(−x) )/2)+∫_x ^∞ ((cos(w)e^(−w) )/w)dw]= (1/2)+lim_(x→0^+ ) [e^(−x) Ci(x)+∫_x ^∞ ((cos(w)e^(−w) )/w)dw]= (1/2)+lim_(x→0^+ ) [e^(−x) Ci(x)+∫_x ^∞ Ci′(w)e^(−w) dw]= (1/2)−∫_0 ^∞ e^(−w) Ci(w)dw=(1/2)−∫_0 ^∞ w′e^(−w) Ci(w)dw= (1/2)−∫_0 ^∞ we^(−w) Ci(w)dw+∫_0 ^∞ cos(w)e^(−w) dw 1−∫_0 ^∞ we^(−w) Ci(w)dw⇒∫_0 ^∞ we^(−w) Ci(w)dw=(1/2)](https://www.tinkutara.com/question/Q220633.png)
$$ \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{w}\mathrm{Ci}\left({w}\right){e}^{−{w}} \mathrm{d}{w} \\ $$$$\int{we}^{−{w}} \mathrm{d}{w}=−{we}^{−{w}} −{e}^{−{w}} \\ $$$$\int\mathrm{cos}\left({w}\right){e}^{−{w}} \mathrm{d}{w}=\int\left(\mathrm{sin}\left({w}\right)\right)'{e}^{−{w}} \mathrm{d}{w}= \\ $$$$\mathrm{sin}\left({w}\right){e}^{−{w}} +\int\mathrm{sin}\left({w}\right){e}^{−{w}} \mathrm{d}{w}= \\ $$$$\mathrm{sin}\left({w}\right){e}^{−{w}} −\int\left(\mathrm{cos}\left({w}\right)\right)'{e}^{−{w}} \mathrm{d}{w}= \\ $$$$\mathrm{sin}\left({w}\right){e}^{−{w}} −\mathrm{cos}\left({w}\right){e}^{−{w}} −\int\mathrm{cos}\left({w}\right){e}^{−{w}} \mathrm{d}{w}\Rightarrow \\ $$$$\Rightarrow\int\mathrm{cos}\left({w}\right){e}^{−{w}} \mathrm{d}{w}=\frac{\mathrm{sin}\left({w}\right){e}^{−{w}} −\mathrm{cos}\left({w}\right){e}^{−{w}} }{\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{x}\mathrm{Ci}\left({x}\right)\right]=\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[\frac{\mathrm{Ci}\left({x}\right)}{\frac{\mathrm{1}}{{x}}}\right]\overset{{H}} {=}\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[\frac{\frac{\mathrm{cos}\left({x}\right)}{{x}}}{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}\right]=\mathrm{0} \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{w}\mathrm{Ci}\left({w}\right){e}^{−{w}} \mathrm{d}{w}=−\underset{\mathrm{0}} {\overset{\infty} {\int}}\left({we}^{−{w}} +{e}^{−{w}} \right)'\mathrm{Ci}\left({w}\right)\mathrm{d}{w}= \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{e}^{−{x}} \mathrm{Ci}\left({x}\right)+\underset{{x}} {\overset{\infty} {\int}}\mathrm{cos}\left({w}\right){e}^{−{w}} \mathrm{d}{w}+\underset{{x}} {\overset{\infty} {\int}}\frac{\mathrm{cos}\left({w}\right){e}^{−{w}} }{{w}}\mathrm{d}{w}\right]= \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{e}^{−{x}} \mathrm{Ci}\left({x}\right)−\frac{\mathrm{sin}\left({x}\right){e}^{−{x}} −\mathrm{cos}\left({x}\right){e}^{−{x}} }{\mathrm{2}}+\underset{{x}} {\overset{\infty} {\int}}\frac{\mathrm{cos}\left({w}\right){e}^{−{w}} }{{w}}\mathrm{d}{w}\right]= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{e}^{−{x}} \mathrm{Ci}\left({x}\right)+\underset{{x}} {\overset{\infty} {\int}}\frac{\mathrm{cos}\left({w}\right){e}^{−{w}} }{{w}}\mathrm{d}{w}\right]= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{e}^{−{x}} \mathrm{Ci}\left({x}\right)+\underset{{x}} {\overset{\infty} {\int}}\mathrm{Ci}'\left({w}\right){e}^{−{w}} \mathrm{d}{w}\right]= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}−\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{w}} \mathrm{Ci}\left({w}\right)\mathrm{d}{w}=\frac{\mathrm{1}}{\mathrm{2}}−\underset{\mathrm{0}} {\overset{\infty} {\int}}{w}'{e}^{−{w}} \mathrm{Ci}\left({w}\right)\mathrm{d}{w}= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}−\underset{\mathrm{0}} {\overset{\infty} {\int}}{we}^{−{w}} \mathrm{Ci}\left({w}\right)\mathrm{d}{w}+\underset{\mathrm{0}} {\overset{\infty} {\int}}\mathrm{cos}\left({w}\right){e}^{−{w}} \mathrm{d}{w} \\ $$$$\mathrm{1}−\underset{\mathrm{0}} {\overset{\infty} {\int}}{we}^{−{w}} \mathrm{Ci}\left({w}\right)\mathrm{d}{w}\Rightarrow\underset{\mathrm{0}} {\overset{\infty} {\int}}{we}^{−{w}} \mathrm{Ci}\left({w}\right)\mathrm{d}{w}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$