作者wuling510665 (摩卡金币)
看板NTUCHE-02-HW
标题Re: [问题] an integral question
时间Sat Nov 14 01:26:23 2009
※ 引述《i563214f (i563214f)》之铭言：
∫ ln(sin(x))dx x from 0 to π/2
please tell me how to calculate this question
thank you
Let I=∫ln(sinx)dx x from 0 to pi/2 ---1
PROPERTY:∫f(x)dx x from 0 to a
=∫f(a-x)dx x from 0 to a
I=∫ln(sin(pi/2-x))dx=∫ln(cosx)dx ---2
1+2:2I=∫(lnsinx+lncosx)dx
=∫ln(sinxcosx)dx
=∫ln(sin2x/2)dx
=∫(ln(sin2x)-ln2)dx
=∫(ln(sin2x)dx-∫(ln2)dx
=∫(ln(sin2x)dx-ln2*x|x from 0 to pi/2
=1/2∫ln(sinu)du-ln2*pi/2 u from 0 to pi
=1/2*2∫ln(sinu)du-ln2*pi/2 (PROPERTY)
=∫ln(sinu)du-ln2*pi/2
=I-ln2*pi/2
2I=I-ln2*pi/2
I=ln2*(-pi/2)
以上= =
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.242.92
1F：推 tp61i6e04 :同学你题目有打错吗? 我画图+计算机+网页积分器 11/14 00:38
2F：→ tp61i6e04 :三种方法算出来都是无解 11/14 00:38
3F：→ ttvic :算不出+1 11/14 00:45
4F：推 tobe6104 :这题不能解阿 11/14 00:57
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.211.173
5F：→ wuling510665:倒数第六行的x from 0 to pi/2 11/14 01:28
6F：→ tp61i6e04 :ln(x) x不能等於0 这篇是.... 11/14 02:12
7F：→ wuling510665:x没有等於0啊!只是从0积到pi/2 11/14 02:15
8F：→ tp61i6e04 :从0积到pi/2不用等於0= =? 11/14 02:22
9F：→ tp61i6e04 :他根本不会形成封闭的区块...哪来的面积 11/14 02:23
10F：→ tp61i6e04 :好ㄅ~我错了 11/14 02:34
11F：推 waanapple :这是有名难题~如果有人能不看解答解出来~真的超强!! 11/14 02:40