Question

求数学大神解答，详细步骤会采纳，谢谢喽

Answer 1

追问

这是根据什么求的？

追答

这是复合函数的求导准则啊！

追问

亲 不忙帮我求一下圈题号的四道题
步骤详细些

追答

(10)
∫[e^x/(1+e^2x)]dx
=∫[1/(1+e^2x)]d(e^x)
=arctan(e^x)+C

(1)
∫[1/√x*(1+x)]dx
=2∫[1/(1+x)]d(√x)
=2arctan(√x)+C

(2)
∫x^2*lnxdx
=(1/3)∫lnxd(x^3)
=(1/3)[x^3*lnx-∫x^3*d(lnx)]
=(1/3)x^3*lnx-(1/3)∫x^3*(1/x)dx
=(1/3)x^3*lnx-(1/3)∫x^2dx
=(1/3)x^3*lnx-(1/9)x^3+C

(3)
∫xsinxdx
=-∫xd(cosx)
=-[x*cosx-∫cosxdx]
=-x*cosx+∫cosxdx
=-x*cosx+sinx+C

Answer 2

泰勒展开：ln(1+x)=x-x^2\2+x^3\3-x^4\4+.......+(-1)^(n-1)x^n\n+O(x^(n+1))

Answer 3

等价无穷小量代换，ln(x+1)≈x，∴=lim(1/x-1/x)=0