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# definite..integration by parts

[2,0]_∫xe^(2x) dx

answer: 1/4 (3e^4 + 1)

### 2 Answers

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- linchLv 71 decade agoFavorite Answer
∫xe^(2x) dx, Let u = x, dv = e^(2x) dx ==> du = dx, v = e^(2x) / 2

= x e^(2x) / 2 - ∫e^(2x) / 2 dx

= x e^(2x) / 2 - e^(2x) / 4 + C

代上限 2 下限 0

( 2 e^4/2 - e^4/4 ) - ( 0 - 1/4) = (3e^4 + 1) / 4

- 1 decade ago
let u= x dv= e^2x dx

du= dx v=1/2*e^2x

原式 = uv-∫v du

= [ 1/2*x*e^2x ]上界2，下界0 ─ 1/2∫e^2x dx

={ 1/2*x*e^2x - 1/2 *1/2 *e^2x } [2,0]帶入上下界 (可以先把不定積分的結果寫下後一併再帶入上下界)

=[ e^4 - 1/4*e^4 ] ─ [ 0-1/4*e^0 ]

=3/4*e^4 + 1/4

=1/4 (3e^4 + 1)

因為無法用方程式編輯器，所以表達出來很奇怪

希望你能看得懂^ ^

Source(s): 我自己

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