汪汪 asked in 科學數學 · 1 decade ago

微積分的improper integrals的問題3?

Which of the following integrals are improper ?why ?

1.∫1/(2x-1)dx 此為定積分上 2下1

2.∫1/(2x-1)dx 此為定積分上 1下0

3.∫sin x/(1+x^2) dx 此為定積分上∞下 -∞

4.∫ln(x-1)dx 此為定積分上 2下1

Update:

convergent和divergent怎麼分?

Update 2:

improper integrals是暇積分?

Update 3:

什麻是暇積分呢?

Update 4:

可以用中文簡單解釋嗎?

Update 5:

∫1/x^2 dx 定積分上∞下1-----converge

∫1/x dx 定積分上∞下1-----diverge

為何呢?

Update 6:

第2題和第4題

都沒有∞的定積分

為何是暇積分?

3 Answers

Rating
  • 1 decade ago
    Favorite Answer

    By the definition of improper integra l:

    An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.

    Hence 1 is general integral, 2,3,4 is improper integral.

    Noting that 2 is improper at x=1/2, 3is improper at x = ∞, -∞ and 4 is improper at x=1.

    圖片參考:http://i286.photobucket.com/albums/ll87/hermittion...

    2009-03-18 14:59:31 補充:

    ∫1/x^2 dx 定積分上∞下1-----converge

    ∫1/x dx 定積分上∞下1-----diverge

    為何呢?

    ∫[1~∞] 1/(x^p) dx

    p>1時收斂,p≤1發散! (微積分教科書就有)

    第2題和第4題

    都沒有∞的定積分

    為何是暇積分?

    請您注意第2題因為被積分函數在1/2處不連續(因為現在積分的上下限為[0~1] , 1/2∈[0~1]所以為瑕積分(improper integral)

    至於第4題 ln(x-1) 因為真數x-1>所以x>1

    顯然x=1這個點也是瑕點喔

  • linch
    Lv 7
    1 decade ago

    improper integrals是暇積分? 是的!!

    2,3,4 都是 improper integral

    其中 2,4 發散 3 收斂

  • 1 decade ago

    只有3,4是暇積分

    2009-03-16 22:00:45 補充:

    2也是

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