✨Danh sách tích phân với hàm lôgarít

Dưới đây là danh sách tích phân với hàm lôgarít.

Chú ý: bài này quy ước x > 0.

: $\backslash int\backslash ln\; cx\backslash ,dx\; =\; x\backslash ln\; cx\; -\; x$ : $\backslash int\; (\backslash ln\; x)^2\backslash ;\; dx\; =\; x(\backslash ln\; x)^2\; -\; 2x\backslash ln\; x\; +\; 2x$ : $\backslash int\; (\backslash ln\; cx)^n\backslash ;\; dx\; =\; x(\backslash ln\; cx)^n\; -\; n\backslash int\; (\backslash ln\; cx)^\{n-1\}\; dx$ : $\backslash int\; \backslash frac\{dx\}\{\backslash ln\; x\}\; =\; \backslash ln|\backslash ln\; x|\; +\; \backslash ln\; x\; +\; \backslash sum^\backslash infty${i=2}\frac{(\ln x)^i}{i\cdot i!} : $\backslash int\; \backslash frac\{dx\}\{(\backslash ln\; x)^n\}\; =\; -\backslash frac\{x\}\{(n-1)(\backslash ln\; x)^\{n-1\; +\; \backslash frac\{1\}\{n-1\}\backslash int\backslash frac\{dx\}\{(\backslash ln\; x)^\{n-1\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$ : $\backslash int\; x^m\backslash ln\; x\backslash ;dx\; =\; x^\{m+1\}\backslash left(\backslash frac\{\backslash ln\; x\}\{m+1\}-\backslash frac\{1\}\{(m+1)^2\}\backslash right)\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; -1\backslash mbox\{)\}$ : $\backslash int\; x^m\; (\backslash ln\; x)^n\backslash ;\; dx\; =\; \backslash frac\{x^\{m+1\}(\backslash ln\; x)^n\}\{m+1\}\; -\; \backslash frac\{n\}\{m+1\}\backslash int\; x^m\; (\backslash ln\; x)^\{n-1\}\; dx\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; -1\backslash mbox\{)\}$ : $\backslash int\; \backslash frac\{(\backslash ln\; x)^n\backslash ;\; dx\}\{x\}\; =\; \backslash frac\{(\backslash ln\; x)^\{n+1\{n+1\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; -1\backslash mbox\{)\}$ : $\backslash int\; \backslash frac\{\backslash ln\; x\backslash ,dx\}\{x^m\}\; =\; -\backslash frac\{\backslash ln\; x\}\{(m-1)x^\{m-1-\backslash frac\{1\}\{(m-1)^2\; x^\{m-1\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; 1\backslash mbox\{)\}$ : $\backslash int\; \backslash frac\{(\backslash ln\; x)^n\backslash ;\; dx\}\{x^m\}\; =\; -\backslash frac\{(\backslash ln\; x)^n\}\{(m-1)x^\{m-1\; +\; \backslash frac\{n\}\{m-1\}\backslash int\backslash frac\{(\backslash ln\; x)^\{n-1\}\; dx\}\{x^m\}\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; 1\backslash mbox\{)\}$ : $\backslash int\; \backslash frac\{x^m\backslash ;\; dx\}\{(\backslash ln\; x)^n\}\; =\; -\backslash frac\{x^\{m+1\{(n-1)(\backslash ln\; x)^\{n-1\; +\; \backslash frac\{m+1\}\{n-1\}\backslash int\backslash frac\{x^m\; dx\}\{(\backslash ln\; x)^\{n-1\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$ : $\backslash int\; \backslash frac\{dx\}\{x\backslash ln\; x\}\; =\; \backslash ln|\backslash ln\; x|$ *: $\backslash int\; \backslash frac\{dx\}\{x^n\backslash ln\; x\}\; =\; \backslash ln|\backslash ln\; x|\; +\; \backslash sum^\backslash infty${i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!} : $\backslash int\; \backslash frac\{dx\}\{x\; (\backslash ln\; x)^n\}\; =\; -\backslash frac\{1\}\{(n-1)(\backslash ln\; x)^\{n-1\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$ : $\backslash int\; \backslash sin\; (\backslash ln\; x)\backslash ;dx\; =\; \backslash frac\{x\}\{2\}(\backslash sin\; (\backslash ln\; x)\; -\; \backslash cos\; (\backslash ln\; x))$ *: $\backslash int\; \backslash cos\; (\backslash ln\; x)\backslash ;dx\; =\; \backslash frac\{x\}\{2\}(\backslash sin\; (\backslash ln\; x)\; +\; \backslash cos\; (\backslash ln\; x))$