推 ccccc7784:請問第二題的第三部要怎麼得來？ 10/29 10:45
→ ejialan:部分分式 用heaviside cover-up method 10/29 12:35
推 ccccc7784:第一題也太難... 10/30 02:09
→ endlesschaos:第一題你就算令 2x - x^2 = u^2 也積得出來 10/30 03:34
※ 引述《ccccc7784 (龍王號)》之銘言：
: 請問這兩題應該怎麼做?
: http://i.imgur.com/K4xS9R6.png
: 是用變數變換嗎?
: 感謝
x^2
∫---------------- dx
(2x - x^2)^0.5
x^2
= ∫------------------- dx
[1 - (x-1)^2]^0.5
let x-1 = sint
dx = cost dt
(sint + 1)^2
= ∫-------------- cost dt
cost
1-cos2t
= ∫--------- dt + 2∫sint dt + ∫dt
2
3 1
= ---t - 2cost - ---sin2t + c
2 4
3 -1 2 0.5 1 2 0.5
= ---sin (x-1) - 2[1 - (x-1) ] - ---(x-1)[1 - (x-1) ] + c,
2 2
where c is an arbitrary constant.
x+3
∫----------- dx
2x^3 - 8x
1 x+3
= ---∫------------- dx
2 x(x+2)(x-2)
1 3 1 1 1 5 1
= ---{∫[(- ---)--- + --- ----- + --- -----] dx}
2 4 x 8 x+2 8 x-2
1 5 3
= ----㏑|x+2| + ----㏑|x-2| - ---㏑|x| + c, where c is an arbitrary constant.
16 16 8
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