[微積] (cscx)^2積分？

看板Math作者 (天蔚藍)時間14年前 (2011/04/14 23:25), 編輯推噓5(5推 0噓 25→)
留言30則, 4人參與, 最新討論串1/2 (看更多)
http://tinyurl.com/3jovf3j 請問這樣算有錯嗎？ 感覺有點怪怪的，最後X是併到C裡面？ -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 118.171.225.45
推
04/14 23:27, , 1F
sinx=u 那麼cosx跟u就有關係啦
04/14 23:27, 1F
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04/14 23:28, , 2F
果然.....
04/14 23:28, 2F
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04/14 23:28, , 3F
第四個等號出問題
04/14 23:28, 3F
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04/14 23:28, , 4F
不過我是令成(dsinx)...只對dsinx去積分
04/14 23:28, 4F
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04/14 23:29, , 5F
這樣還有關？
04/14 23:29, 5F
推
04/14 23:29, , 6F
對
04/14 23:29, 6F
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04/14 23:30, , 7F
okay 那 cos^2/sin^2 有什麼積分方法比較好?
04/14 23:30, 7F
推
04/14 23:33, , 8F
還記得cotx的微分嗎?
04/14 23:33, 8F
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04/14 23:33, , 9F
可以直接用...
04/14 23:33, 9F
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04/14 23:33, , 10F
ln sin 我用用看 感謝~
04/14 23:33, 10F
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04/14 23:35, , 11F
囧技成積分的去了
04/14 23:35, 11F
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04/14 23:35, , 12F
cotx的微分我會阿，我想把csc^2推回去cotx ~
04/14 23:35, 12F
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04/14 23:55, , 13F
04/14 23:55, 13F
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04/14 23:55, , 14F
這樣應該沒錯了吧?!
04/14 23:55, 14F
推
04/15 00:15, , 15F
可以在c/(s^2) ds的時候就用分部積分
04/15 00:15, 15F
推
04/15 00:49, , 16F
Let t = csc x + cot x, then we have
04/15 00:49, 16F
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04/15 00:50, , 17F
(1) dt/t = csc x dx. (2) (t + 1/t)/2 = csc x.
04/15 00:50, 17F
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04/15 00:51, , 18F
Using above, we can calculate S (csc x)^n dx for
04/15 00:51, 18F
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04/15 00:51, , 19F
any positive integer n, and similarly for secx.
04/15 00:51, 19F
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04/15 00:52, , 20F
(1) is wrong, and (1) should be dt/t = - csc x dx
04/15 00:52, 20F
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04/15 00:54, , 21F
For example, S (csc x)^2 dx = S csc x (csc x) dx
04/15 00:54, 21F
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04/15 00:54, , 22F
= -1/2 S 1 + t^(-2) dt = ... .
04/15 00:54, 22F
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04/15 01:10, , 23F
BUT there's some problem
04/15 01:10, 23F
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04/15 01:10, , 24F
last i get -1/2[t-1/t]+c
04/15 01:10, 24F
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04/15 01:11, , 25F
BUT t+1/t seems equal to zero ?
04/15 01:11, 25F
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04/15 01:12, , 26F
=================================================
04/15 01:12, 26F
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04/15 01:13, , 27F
(csc+cot)-1/(csc+cot) = (csc^2-cot^2-1)/(csc+cot)
04/15 01:13, 27F
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04/15 01:14, , 28F
and we know 1+cot^2=csc^2 so csc^2-cot^2-1 = 0
04/15 01:14, 28F
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04/15 01:16, , 29F
SORRY MY FAULT ~
04/15 01:16, 29F
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04/15 01:17, , 30F
I GET IT THANK YOU VERY MUCH
04/15 01:17, 30F
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文章代碼(AID): #1Dfn7kOg (Math)
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文章代碼(AID): #1Dfn7kOg (Math)