J-0-e-t-u-pi-t-cos-t-dt-note-u-c-t-0-t-lt-c-1-t-gt-c-c-0-

Question Number 220842 by mnjuly1970 last updated on 20/May/25

J=∫_0 ^( ∞) e^(−t) u_π (t)cos(t)dt=? note: u_c (t)= { (( 0 t<c)),(( 1 t>c )) :} ; c≥0

$$ \\ $$$$\:\:\:\:\:\:\mathrm{J}=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{t}} {u}_{\pi} \left({t}\right){cos}\left({t}\right){dt}=? \\ $$$$ \\ $$$$\:{note}:\:\:{u}_{{c}} \left({t}\right)=\:\begin{cases}{\:\mathrm{0}\:\:\:\:\:\:\:\:{t}<{c}}\\{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{t}>{c}\:\:}\end{cases}\:\:;\:\:\:\:{c}\geqslant\mathrm{0} \\ $$

Commented by SdC355 last updated on 20/May/25

∫_0 ^( ∞) e^(−kt) u_π (t)cos(t)dt ∫_0 ^( π) e^(−kt) cos(t)dt+∫_π ^( ∞) e^(−t) cos(t)dt ∫ e^(−w) cos(w)dw=(1/2)e^(−w) (sin(w)−cos(w))+C ∫_π ^( ∞) =−(1/2)e^(−π)

$$\int_{\mathrm{0}} ^{\:\infty} \:{e}^{−{kt}} {u}_{\pi} \left({t}\right)\mathrm{cos}\left({t}\right)\mathrm{d}{t} \\ $$$$\int_{\mathrm{0}} ^{\:\pi} \:{e}^{−{kt}} \mathrm{cos}\left({t}\right)\mathrm{d}{t}+\int_{\pi} ^{\:\infty} \:{e}^{−{t}} \mathrm{cos}\left({t}\right)\mathrm{d}{t} \\ $$$$\int\:{e}^{−{w}} \mathrm{cos}\left({w}\right)\mathrm{d}{w}=\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{w}} \left(\mathrm{sin}\left({w}\right)−\mathrm{cos}\left({w}\right)\right)+{C} \\ $$$$\int_{\pi} ^{\:\infty} \:=−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\pi} \\ $$

Commented by mnjuly1970 last updated on 20/May/25

$${thanks}\:{alot}\:{sir}… \\ $$

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J-0-e-t-u-pi-t-cos-t-dt-note-u-c-t-0-t-lt-c-1-t-gt-c-c-0-

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