Lv 4
asked in 科學數學 · 1 decade ago

definite..integration by parts

[2,0]_∫xe^(2x) dx

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

2 Answers

Rating
  • linch
    Lv 7
    1 decade ago
    Favorite 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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