的宜 發問時間: 科學數學 · 7 年前

緊急!!微積分的逐次積分 !!20點!!

https://www.facebook.com/photo.php?fbid=6134488486...

以上網址是題目的照片

照片中中有3題請幫忙解一下

20點立刻送上!!

需要詳解喔

1 個解答

評分
  • Tony
    Lv 4
    7 年前
    最佳解答

    這是我的標記, 這是不正式, 請不要在其他地方使用

    (2∫1)(x)dx=積分x由1到2

    2[x]1=(2-1)

    解釋: (2∫1)(xy)dx

    因為dx是對象x, 所以可以先把y當成常數,

    (2∫1)(xy)dx = 2[(1/2)(x^2)(y)]1

    然後2和1是作用x身上,

    所以[(1/2)(2^2)(y) - (1/2)(1^2)(y)]

    1. (ln2∫0)[(y∫0)(e)^(3x+y)dx]dy

    =(ln2∫0) {y[(1/3)(e)^(3x+y)]0} dy

    =(ln2∫0) [(1/3)(e)^(3y+y) - (1/3)(e)^(3(0)+y)] dy

    =(ln2∫0) [(1/3)(e)^(4y) - (1/3)(e)^(y)] dy

    =ln2[(1/12)(e)^(4y)]0 - ln2[(1/3)(e)^(y)]0

    =[(1/12)(e)^(4ln2) - (1/12)(e)^(4(0))]

    - [(1/3)(e)^(ln2) - [(1/3)(e)^(0)]

    =[(1/12)(e)^(ln2^4) - (1/12)] - [(1/3)(e)^(ln2) - (1/3)]

    =[(1/12)(2^4) - (1/12)] - [(1/3)(2) - (1/3)]

    =16/12 - 1/12 - 2/3 + 1/3

    =11/12

    2. (1∫0)[(x^2∫x)(2x+xy)dy]dx

    =(1∫0) {x^2[(2xy+(1/2)(xy^2)]x} dx

    =(1∫0) { [(2x)(x^2)+(1/2)(x)(x^2)^2] - [(2x)(x)+(1/2)(x)(x)^2] } dx

    =(1∫0) { [(2x^3)+(1/2)(x^5)] - [(2x^2)+(1/2)(x^3)] } dx

    =(1∫0) [ (2x^3)+(1/2)(x^5) - (2x^2) - (1/2)(x^3) ] dx

    =(1∫0) [ (1/2)(x^5) + (3/2)(x^3) - (2x^2) ] dx

    =1[(1/12)(x^6) + (3/8)(x^4) - (2/3)(x^3)]0 dx

    =[(1/12)(1^6) + (3/8)(1^4) - (2/3)(1^3)]

    - [(1/12)(0^6) + (3/8)(0^4) - (2/3)(0^3)]

    =(1/12) + (3/8) - (2/3)

    = -5/24

    3. (e∫1)[(x∫0)(lnx)dy]dx

    =(e∫1) {x[(lnx)(y)]0} dx

    =(e∫1) [(lnx)(x) - (lnx)(0)] dx

    =(e∫1)[(lnx)(x)]dx

    (dx^2)/dx = 2x

    => 2xdx=dx^2

    => xdx=(1/2)dx^2

    ∫ [(lnx)(x)]dx

    =(1/2) ∫ [(lnx)]d(x^2)

    =(1/2)(x^2)(lnx) - (1/2) ∫ [(x^2)]d(lnx)

    =(1/2)(x^2)(lnx) - (1/2) ∫ [(x^2)(1/x)]dx

    =(1/2)(x^2)(lnx) - (1/2) ∫ [(x)]dx

    =(1/2)(x^2)(lnx) - (1/4)(x^2)

    所以,(e∫1)[(x∫0)(lnx)dy]dx

    =(e∫1)[(lnx)(x)]dx

    =e[(1/2)(x^2)(lnx) - (1/4)(x^2)]1

    =[(1/2)(e^2)(lne) - (1/4)(e^2)] - [(1/2)(e^2)(ln1) - (1/4)(1^2)]

    =[(1/2)(e^2) - (1/4)(e^2)] - [0 - (1/4)]

    =(1/4)(e^2) + (1/4)

    =(1/4)(e^2 + 1)

    積分的步驟好容易出錯, 但現在你應該知道怎樣計算吧

    因為我是香港人, 所以可能打的中文數學用語有點奇怪(我用英文學的)

    希望你看得懂.

    希望幫到你!

    參考資料: 自己
還有問題?馬上發問,尋求解答。