definite integration

不是嗎，myisland8132，計算∫ ln(1+x^3) dx也要淪落到用數值積分？
http://integrals.wolfram.com/index.jsp?expr=ln%281%2Bx%5E3%29&random=false

2010-12-18 02:40:00 補充：

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/limit/limitusingriemannsum9.jpg

參考資料：
my maths knowledge

∫ ln(1+x^3) dx [from 0 to 1] = 0.200094 not -2.8

lim (n->infinity) {(n^3+1)(n^3+2^3)(n^3+3^3)...(n^3+n^3) / n^(3n)}^(1/n)
=lim (n->infinity) {(1+(1/n)^3)(1+(2/n)^3)(1+(3/n)^3)...(1+(n/n)^3)}^(1/n)

Now let y=lim (n->infinity) {(1+(1/n)^3)(1+(2/n)^3)(1+(3/n)^3)...(1+(n/n)^3)}^(1/n), then lny
=lim (n->infinity) (1/n) {ln(1+(1/n)^3)+ln(1+(2/n)^3)+ln(1+(3/n)^3)...ln(1+(n/n)^3)}
= ∫ ln(1+x^3) dx [from 0 to 1]
=xln(1+x^3)|[0,1]- ∫ 3x^3/(1+x^3) dx [from 0 to 1]
=ln2-3 ∫ x^3/(1+x^3) dx [from 0 to 1]
There is no closed form of the last integral. By approximation, its value is around 1.16435. So, lny=-2.8 and the value of the original expression is e^(-2.8) = 0.06082