题目

求不定积分 int dfrac (1)({x)^3}ln xdx

$求不定积分\int \frac{1} {x^3} \ln xdx$

题目解答

答案

$\begin{aligned}&\\ &\\ & \int \frac{1}{x^3}\ln x\,dx\\ & =\int x^{-3}\ln x\,dx\\ & \\ &使用分部积分法，令 u = \ln x，dv = x^{-3}\,dx\\ &则 du = \frac{1}{x}\,dx，v = -\frac{1}{2}x^{-2}\\ & \int x^{-3}\ln x\,dx\\ & =-\frac{1}{2}x^{-2}\ln x - \int (-\frac{1}{2}x^{-2})\frac{1}{x}\,dx\\ & =-\frac{1}{2}x^{-2}\ln x + \frac{1}{2}\int x^{-3}\,dx\\ & =-\frac{1}{2}x^{-2}\ln x + \frac{1}{2}(-\frac{1}{2}x^{-2}) + C\\ & =-\frac{1}{2}x^{-2}\ln x - \frac{1}{4}x^{-2} + C\\ & =-\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C\\ & \\ &\\ & \\&\end{aligned}$