formulario sugli integrali

INTEGRALI

PRIMITIVA DI UNA FUNZIONE

E’ una funzione \(F\left( x \right)\)tale che \(F’\left( x \right)=f\left( x \right)\)

INTEGRALI INDEFINITI

\(\int{f\left( x \right)dx}=F\left( x \right)+c\)

INDEGRALI DEFINITI

\(\int\limits_{a}^{b}{f\left( x \right)dx}=F\left( b \right)-F\left( a \right)\)

INTEGRALI IMMEDIATI DELLE FUNZIONI FONDAMENTALI

\(\int{{{x}^{\alpha }}dx}=\frac{{{x}^{\alpha +1}}}{\alpha +1}\,\,+c\,\,con\,\,\,\alpha \ne -1\)

\(\int{\frac{1}{x}dx}=\ln \left| x \right|+c\)

\(\int{{{e}^{x}}dx}={{e}^{x}}+c\)

\(\int{\sin x\,dx\,}=-\cos x\,+\,c\)

\(\int{\cos x\,dx\,}=\sin x\,+\,c\)

\(\int{\frac{1}{{{\cos }^{2}}x}\,dx\,}=\int{\left( 1+{{\tan }^{2}}x \right)dx\,}=\tan x\,+\,c\)

\(\int{\frac{1}{{{\sin }^{2}}x}\,dx\,}=\int{\left( 1+{{\cot }^{2}}x \right)\,dx\,}=-\cot x\,+\,c\)

\(\int{\frac{1}{1+{{x}^{2}}}dx}=\arctan x+c\)

\(\int{\frac{1}{\sqrt{1-{{x}^{2}}}}dx}=\arcsin x+c\)

INTEGRALI LA CUI PRIMITIVA E’ UNA FUNZIONE COMPOSTA

\(\int{{{\left[ f\left( x \right) \right]}^{\alpha }}{f}’\left( x \right)dx}=\frac{{{\left[ f\left( x \right) \right]}^{\alpha +1}}}{\alpha +1}\,\,+c\,\,con\,\,\,\alpha \ne -1\)

\(\int{\frac{{f}’\left( x \right)}{f\left( x \right)}dx}=\ln \left| f\left( x \right) \right|+c\)

\(\int{{{e}^{f\left( x \right)}}{f}’\left( x \right)dx}={{e}^{f\left( x \right)}}+c\)

\(\int{\sin \left[ f\left( x \right) \right]\,{f}’\left( x \right)dx\,}=-\cos x\,+\,c\)

\(\int{\cos \left[ f\left( x \right) \right]\,{f}’\left( x \right)dx\,}=\sin x\,+\,c\)

\(\int{\frac{{f}’\left( x \right)}{{{\cos }^{2}}f\left( x \right)}\,dx\,}=\int{\left( 1+{{\tan }^{2}}\left[ f\left( x \right) \right] \right){f}’\left( x \right)dx\,}=\tan f\left( x \right)\,+\,c\)

\(\int{\frac{{f}’\left( x \right)}{{{\sin }^{2}}\left[ f\left( x \right) \right]}\,dx\,}=\int{\left( 1+{{\cot }^{2}}\left[ f\left( x \right) \right] \right)\,{f}’\left( x \right)dx\,}=-\cot f\left( x \right)\,+\,c\)

\(\int{\frac{f\left( x \right)}{1+f{{\left( x \right)}^{2}}}dx}=\arctan f\left( x \right)+c\)

\(\int{\frac{{f}’\left( x \right)}{\sqrt{1-{{\left[ f\left( x \right) \right]}^{2}}}}dx}=\arcsin x+c\)