Cet article donne les primitives de fonctions logarithmes. ∀ n ∈ Z ∖ { − 1 } , ∫ ( ln x ) n d x x = ( ln x ) n + 1 n + 1 + C {\displaystyle \forall n\in \mathbb {Z} \setminus \{-1\},\quad \int (\ln x)^{n}~{\frac {\mathrm {d} x}{x}}={\frac {(\ln x)^{n+1}}{n+1}}+C} ∀ m ∈ Z ∖ { − 1 } , ∀ n ∈ N , ∫ ( ln x ) n x m d x = x m + 1 ∑ k = 0 n ( − 1 ) n − k n ! ( m + 1 ) n − k + 1 k ! ( ln x ) k + C {\displaystyle \forall m\in \mathbb {Z} \setminus \{-1\},\forall n\in \mathbb {N} ,\quad \int (\ln x)^{n}x^{m}~\mathrm {d} x=x^{m+1}\sum _{k=0}^{n}{\frac {(-1)^{n-k}n!}{(m+1)^{n-k+1}k!}}(\ln x)^{k}+C} ∀ m ∈ Z , {\displaystyle \forall m\in \mathbb {Z} ,} ∀ n ∈ { 2 , 3 , … } , ∫ x m ( ln x ) n d x = − x m + 1 ∑ k = 1 n − 1 ( k − 1 ) ! ( m + 1 ) n − 1 − k ( n − 1 ) ! ( ln x ) k + ( m + 1 ) n − 1 ( n − 1 ) ! ∫ x m ln x d x + C {\displaystyle \forall n\in \{2,3,\ldots \},\quad \int {\frac {x^{m}}{(\ln x)^{n}}}~\mathrm {d} x=-x^{m+1}\sum _{k=1}^{n-1}{\frac {(k-1)!(m+1)^{n-1-k}}{(n-1)!(\ln x)^{k}}}+{\frac {(m+1)^{n-1}}{(n-1)!}}\int {\frac {x^{m}}{\ln x}}~\mathrm {d} x+C} et ∫ x m ln x d x = ln | ln x | + ∑ n = 1 + ∞ ( m + 1 ) n ( ln x ) n n ! ⋅ n + C {\displaystyle {\text{et}}\quad \int {\frac {x^{m}}{\ln \,x}}~\mathrm {d} x=\ln |\ln \,x|+\sum _{n=1}^{+\infty }{\frac {(m+1)^{n}(\ln \,x)^{n}}{n!\cdot n}}+C} ∀ a ≠ 0 , ∫ e a x ln x d x = 1 a e a x ln | x | − 1 a ∫ e a x x d x {\displaystyle \forall a\neq 0,\quad \int \mathrm {e} ^{ax}\ln x~\mathrm {d} x={\frac {1}{a}}\mathrm {e} ^{ax}\ln |x|-{\frac {1}{a}}\int {\frac {\mathrm {e} ^{ax}}{x}}~\mathrm {d} x} Article principal : Primitive. Cet article est une ébauche concernant l’analyse. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. (en) Milton Abramowitz et Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [détail de l’édition] (lire en ligne) Portail de l'analyse Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for Firefox
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.