M1數學題一問?10點

👨🏻‍🏫 學術台數學

wai1994926

2012-03-06 4:17 am

/(1/v + 1/(2-v))dv

點變lnv-ln(2-v) + C

更新1:

why ʃ 1/(2-v) = ln(2-v)(-1) +C

更新2:

ʃ 1/(2-v) = ln(2-v)(-1) +C (-1)點出

回答 (3)

用戶10012

2012-03-06 3:56 pm

✔ 最佳答案

For ∫ dv/(2 -v).
By Substitution method, let 2 - v = u, so d(2 - v) = du = - dv. Or dv = - du.
So the integral becomes ∫ - du/u = - ln u = = - ln ( 2 - v) + C.

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用戶25975

2012-03-06 4:47 pm

Note that the most proper ans should not miss the absolute sign, i.e.

ʃdx/x = ln |x| + C

So for ʃdv/v, unless we are sure that v is positive, the ans should be ln |v| + C

Similar judgement for ʃdv/(2 - v)

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TwoThounsands

2012-03-06 4:28 am

ʃ (1/v + 1/(2-v))dv

= lnv-ln(2-v) + C

ʃ 1/x dx = lnx +C

ʃ 1/(2-v) = ln(2-v)(-1) +C

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