查看完整版本: (定積分)數學題
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op89q1 發表於 2012-5-24 10:25 AM

(定積分)數學題

計算以下積分
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op89q1 發表於 2012-5-24 10:27 AM

對不起,忘了貼圖   


bjl1212 發表於 2012-5-24 08:16 PM

這個題目是微積分的嗎,太難了吧
我做了第一題,有趣
1.\int_0^1 e^{-x}sin(x)/x dx+\int_1^(infinite)e^{-x}sin(x)/x dx (這部分是瑕積分 用integration by part 可以積不過
   \int_0^1 e^{-x}sin(x)/x dx>\int_0^1 sin(1)/x dx=sin(1)\int_0^1 1/x dx 發散
    這裡lim_{x->0}sin(x)/x =1 所以不是瑕積分

bjl1212 發表於 2012-5-24 08:24 PM

3.和1類似 \int_0^1 {arctan(pix)-arctan(x)}/x dx>\int_0^1{arctan(pi)-arctan(1)}/x
   瑕積分發散
\int_0^(infinite) {arctan(pix)-arctan(x)}/x dx<\int_0^(infinite)1/x dx

bjl1212 發表於 2012-5-24 08:36 PM

2. 0是瑕積分 1不是
(x-1)(ln(x)) 式遞增函數,而且>=0所以
\int_0^1(x-1)/(ln(x)) dx>=\int_0^1 -1/(ln(x)) dx  發散
太難了<br><br><br><br><br><div></div>

op89q1 發表於 2012-5-25 10:18 AM

這個題目應該都不是發散的
我看了書後面的答案
第一題是pi/4
第二題是ln2
第三題是(pi/4)lnpi
不過還是不懂怎樣做

bjl1212 發表於 2012-5-25 11:31 AM

把作法放上來,增點見識

op89q1 發表於 2012-5-25 12:17 PM

我也想知道,不過書裡只有問題跟答案,沒有作法

怪人exe 發表於 2012-5-26 12:41 AM

好樣的...那只能等其他高手或神人了
我也想看看怎麼解

bjl1212 發表於 2012-5-26 08:13 PM

的確 我算錯了瑕積分是收斂,可是要找到closed from???我很懷疑!估計值不難<br><br><br><br><br><div></div>

huiken2005 發表於 2012-5-28 04:13 AM

我看了一看,第一條應該by part兩次就完了,product rule的一部份會cancel呢...
第二條沒頭緒
第三條可以試試分成兩個不同的integral做代入法,三角等式不太記得了...

今天看看有沒有時間解一下...

super6358 發表於 2012-7-24 05:51 AM

本帖最後由 super6358 於 2012-7-24 05:52 AM 編輯

第2題可以拆成雙重積分,這題很技巧,沒碰過還蠻難想到的,要先知道誰的積分會變成〈x-1〉/lnx,答案是  
積分x^ydy上下限:0至1所已重新整理會變成:雙重積分x^ydydx〈上下限,
外0-1內0-1:此時要掉換積分次序,改成先積x,所以會變成:積分x^〈y+1〉/y+1 dy代上下限x=0-1變成:積分1/〈y+1〉dy 上下限0-1變成ln2
所以答案是ln2
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