假设x变量n∈R.
要由导数始推导:
考虑函数y = xⁿ
则y' = nxⁿ⁻¹
(xⁿ)'_x
= lim(Δx->0) [ (x+Δx)ⁿ - xⁿ ]/Δx运用二项式定理展
= lim(Δx->0) [ (xⁿ+nxⁿ⁻¹Δx+O(Δx)) - xⁿ ]/Δx
= lim(Δx->0) (nxⁿ⁻¹Δx+O(Δx))/Δx
= lim(Δx->0) [ nxⁿ⁻¹+O(Δx) ]O(Δx)比Δx更高阶项
= nxⁿ⁻¹
n替换n+1
即(xⁿ⁺¹)'_x = (n+1)xⁿ
即[xⁿ⁺¹/(n+1)]'_x = xⁿ
所两边取定积
∫ xⁿ dx = xⁿ⁺¹/(n+1) + CC任意数项
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