Re: [微積] 分部積分
※ 引述《rebe212296 (綠豆冰)》之銘言：
: 求∫t^4*e^tdt
: 請問怎麼解 謝謝!
t
令 u = t^4 , dv = e dt
則 du = (4)(t^3) dt , v = e^t
∫(t^4)(e^t) dt
= (t^4)(e^t) - (4)(∫(t^3)(e^t) dt)
= (t^4)(e^t) - (4)((t^3)(e^t) - ∫(e^t)((3)(t^2)) dt)
(令 u = t^3 , dv = e^t dt , 則 du = (3)(t^2) dt , v = e^t)
= (t^4)(e^t) - (4)(t^3)(e^t) + 12∫(t^2)(e^t) dt
= (t^4)(e^t) - (4)(t^3)(e^t) + (12)((t^2)(e^t) - ∫(e^t)(2t) dt)
(令 u = t^2 , dv = e^t dt , 則 du = 2t dt , v = e^t)
= (t^4)(e^t) - (4)(t^3)(e^t) + (12)(t^2)(e^t) - 24∫(t)(e^t) dt)
= (t^4)(e^t)-(4)(t^3)(e^t)+(12)(t^2)(e^t)-(24)((t)(e^t) - ∫e^t dt)
(令 u = t , dv = e^t dt , 則 du = dt , v = e^t)
= (t^4)(e^t)-(4)(t^3)(e^t)+(12)(t^2)(e^t)-(24)(t)(e^t) + (24)(e^t) + c
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