(a) y=ln(1+e^x), consider dydx, evaluate in(0 to1) (e^x)/(1+e^x) dx
(b) hence evaluate in(0 to1) 1/(1+e^x) dx
the anwer for (a) should be ln((1+e)/2)) and (b) should be ln((2e)/(1+e))
definite integral please help
2011-11-29 12:04 am
回答 (1)
2011-11-29 12:14 am
✔ 最佳答案
(a) y = ln(1+e^x)dy/dx = 1/(1+e^x) * e^x = e^x/(1+e^x)
∫e^x/(1 + e^x) {0,1} = [ln(1+e^x)] {0,1}
= ln(1+e) - ln 2
= ln[(1+e)/2]
(b) ∫1/(1+e^x) dx {0,1}
= ∫(1+e^x - e^x)/(1+e^x) dx {0,1}
= ∫dx {0,1} - ∫e^x/(1+e^x) dx {0,1}
= [x] {0,1} - ln[(1+e)/2]
= 1 - ln[(1+e)/2]
= ln e - ln[(1+e)/2]
= ln[2e/(1+e)]
參考: Knowledge is power.
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