対数関数の原始関数の一覧
∫ x m d x ( ln  x ) n = − x m + 1 ( n − 1 ) ( ln  x ) n − 1 + m + 1 n − 1 ∫ x m d x ( ln  x ) n − 1 (for n ≠ 1 ) {\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫ d x x ln  x = ln  | ln  x | {\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
∫ d x x n ln  x = ln  | ln  x | + ∑ k = 1 ∞ ( − 1 ) k ( n − 1 ) k ( ln  x ) k k ⋅ k !
∫ ln  a x d x = x ln  a x − x {\displaystyle \int \ln ax\;dx=x\ln ax-x}
∫ ln  ( a x + b ) d x = ( a x + b ) ln  ( a x + b ) − ( a x + b ) a {\displaystyle \int \ln(ax+b)\;dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}}}
∫ ( ln  x ) 2 d x = x ( ln  x ) 2 − 2 x ln  x + 2 x {\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}
∫ ( ln  x ) n d x = x ∑ k = 0 n ( − 1 ) n − k n ! k ! ( ln  x ) k {\displaystyle \int (\ln x)^{n}\;dx=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}
∫ d x ln  x = ln  | ln  x | + ln  x + ∑ k = 2 ∞ ( ln  x ) k k ⋅ k !