対数関数の原始関数の一覧
∫ ⋯ ∫ ⏟ n ln  x d x ⋯ d x ⏟ n = x n n ! ( ln x − ∑ k = 1 n 1 k ) + ∑ k = 0 n − 1 C k x k k ! {\displaystyle \underbrace {{\biggl .}{\biggr .}\int \cdots \int } _{n}\,\ln x\,\underbrace {{\biggl .}{\biggr .}{\,\mathrm {d} }x\cdots {\,\mathrm {d} }x} _{n}={\frac {x^{n}}{n!}}\left(\ln \,x-\sum _{k=1}^{n}{\frac {1}{k}}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}}{k!}}}
∫ x m ln  x d x = x m + 1 ( ln  x m + 1 − 1 ( m + 1 ) 2 ) (for m ≠ − 1 ) {\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}
∫ x m ( ln  x ) n d x = x m + 1 ( ln  x ) n m + 1 − n m + 1 ∫ x m ( ln  x ) n − 1 d x (for m ≠ − 1 ) {\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}
∫ ( ln  x ) n d x x = ( ln  x ) n + 1 n + 1 (for n ≠ − 1 ) {\displaystyle \int {\frac {\left(\ln x\right)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}