不定积分：∫xnlnx dx\int x^n\ln x\,dx∫xnlnxdx Easy integration question 5 专题 General / 综合 难度 L4 来源 QuantQuestion 题目详情 Compute ∫xnln(x)dx.\int x^n \ln (x) dx.∫xnln(x)dx. 解析 对 n≠−1n\ne -1n=−1，分部积分取 u=lnx, dv=xndxu=\ln x,\ dv=x^n dxu=lnx, dv=xndx，则 du=dx/xdu=dx/xdu=dx/x，v=xn+1n+1v=\frac{x^{n+1}}{n+1}v=n+1xn+1： ∫xnlnx dx=xn+1n+1lnx−1n+1∫xndx=xn+1n+1lnx−xn+1(n+1)2+C.\int x^n\ln x\,dx=\frac{x^{n+1}}{n+1}\ln x-\frac{1}{n+1}\int x^n dx =\boxed{\frac{x^{n+1}}{n+1}\ln x-\frac{x^{n+1}}{(n+1)^2}+C}.∫xnlnxdx=n+1xn+1lnx−n+11∫xndx=n+1xn+1lnx−(n+1)2xn+1+C. 若 n=−1n=-1n=−1，则 ∫lnxxdx=12(lnx)2+C\int \frac{\ln x}{x}dx=\frac12(\ln x)^2+C∫xlnxdx=21(lnx)2+C。