WikiDer > Liste der Integrale

Integrieren ist eine grundlegende Bearbeitung der Analyse. Da die Integration nicht wie bei unterscheiden, nach einfachen Regeln sind Tabellen mit gemeinsamen Integralen ein nützliches Werkzeug. Im unten Liste der Integrale besteht aus vielen verschiedenen Funktionen das primitive Funktion gegeben.

Es gibt Listen von Integralen:

• von rationalen Funktionen
• von irrationalen Funktionen
• von Exponentialfunktionen
• von logarithmischen Funktionen
• von trigonometrischen Funktionen
• von inversen trigonometrischen Funktionen
• von hyperbolischen Funktionen
• von inversen hyperbolischen Funktionen

In den hier und in diesen anderen Listen angegebenen Integralen ist C immer eine Integrationskonstante, die nur mit Zusatzinformationen verwendet werden kann, ausgangsbedingung oder Randbedingung, bestimmt werden. Das Primitiv einer Funktion wird bis auf die Integrationskonstante bestimmt.

Mit den hier angegebenen Integralen sind die untere und obere Schranke von Intervall, die integriert ist, ist nicht gegeben. Unechte Integrale werden gesondert behandelt.

Berechnungsregeln beim Integrieren

• Linearität eines Integrals:

${displaystyle int cf(x),{m {d}}x=cint f(x),{m {dx}}}$

${displaystyle int (f(x) g(x)),{m {d}}x=int f(x),{m {d}}x int g(x),{m {d}}x}$

• Teilintegration

${displaystyle int f(x)g'(x),{m {d}}x=f(x)g(x)-int f'(x)g(x),{m {d}}x}$

• bestimmtes Integral

${displaystyle int _{a}^{b}{frac {{m {d}}F(x)}{{m {d}}x}},{m {d}}x=[F(x)] _{a}^{b}=F(b)-F(a)}$

• Mehrfachintegral als iteriertes Integral

${displaystyle iint f(x,y),{m {d}}x{m {d}}y=int left(int f(x,y),{m {d}}xight){m {d}} ich}$

• Integration durch Substitution

${displaystyle int f(g(t)),g'(t),{m {d}}t=int f(x),{m {d}}x}$

Integrationen von Standardfunktionen

Rationale Funktionen

${displaystyle int 1,{m {d}}x=x C}$

${displaystyle int x^{n},{m {d}}x={frac {x^{n 1}}{n 1}} Cqquad {mbox{ as }}neq -1}$

${displaystyle int {frac {1}{x}},{m {d}}x=ln {left|xight|} C}$

${displaystyle int {frac {1}{a^{2} x^{2}}},{m {d}}x={frac {1}{a}}arktan {frac {x}{a}} C }$

${displaystyle int {frac {1}{xleft(a bxight)}},{m {d}}x={frac {1}{a}}ln left|{frac {x}{a bx}}ight| C}$

${displaystyle int {frac {1}{ax^{2} bx c}},{m {d}}x=left{{ egin{matrix}{cfrac {1}{sqrt {b^{2}-4ac} }}ln links|{cfrac {2ax b-{sqrt {b^{2}-4ac}}}{2ax b {sqrt {b^{2}-4ac}}}}ight| C&{mbox{if}} b^{2}>4ac/{cfrac {2}{sqrt {4ac-b^{2}}}}arctan {cfrac {2ax b}{sqrt {4ac-b^{2} }}} C&{mbox{if}} b^{2}<4acend{matrix}}ight.}$

${displaystyle int {frac {x}{ax^{2} bx c}},{m {d}}x={frac {1}{2a}}ln left|ax^{2} bx cight|-{frac {b}{2a}}int {frac {1}{ax^{2} bx c}},{m {d}}x}$

Irrationale Funktionen

${displaystyle int {{m {d}}u über {sqrt {a^{2}-u^{2}}}}=Arcsin {u über a} C}$

${displaystyle int {-{m {d}}u über {sqrt {a^{2}-u^{2}}}}=arccos {u über a} C}$

${displaystyle int {{m {d}}u über u{sqrt {u^{2}-a^{2}}}}={1 über a}{mbox{arcsec}},{|u| über a} C}$

${displaystyle int {sqrt {a^{2}-x^{2}}},{m {d}}x={frac {x}{2}}{sqrt {a^{2}-x^{2 }}} {frac {a^{2}}{2}}Arcsin {frac {x}{a}} C,(a>0)}$

${displaystyle int left(a^{2}-x^{2}ight)^{frac {3}{2}},{m {d}}x={frac {x}{8}}left(5a^ {2}-2x^{2}ight){sqrt {a^{2}-x^{2}}} {frac {3a^{4}}{8}}Arcsin {frac {x}{a}} C,(a>0)}$

${displaystyle int {frac {1}{left(a^{2}-x^{2}ight)^{frac {3}{2}}}},{m {d}}x={frac {x} {a^{2}{sqrt {a^{2}-x^{2}}}}} C}$

${displaystyle int x{sqrt {a bx}},{m {d}}x={frac {2left(3bx-2aight)left(a bxight)^{frac {3}{2}}}{15b^{2 }}}C}$

${displaystyle int {frac {sqrt {a bx}}{x}},{m {d}}x=2{sqrt {a bx}} aint {frac {1}{x{sqrt {a bx}}}} ,{m{d}}x}$

${displaystyle int {frac {x}{sqrt {a bx}}},{m {d}}x={frac {2left(bx-2aight){sqrt {a bx}}}{3b^{2}}} C}$

${displaystyle int {frac {1}{x{sqrt {a bx}}}},{m {d}}x={frac {1}{sqrt {a}}}ln links|{frac {{sqrt {a bx}}-{sqrt {a}}}{{sqrt {a bx}} {sqrt {a}}}}ight| C,(a>0)}$

${displaystyle int {frac {1}{x{sqrt {a bx}}}},{m {d}}x={frac {2}{sqrt {-a}}}arctan links|{sqrt {frac {a bx}{-a}}}acht| C,(a<0)}$

${displaystyle int {frac {sqrt {a^{2}-x^{2}}}{x}},{m {d}}x={sqrt {a^{2}-x^{2}}} -aln links|{frac {a {sqrt {a^{2} x^{2}}}}{x}}ight| C}$

${displaystyle int x{sqrt {a^{2}-x^{2}}},{m {d}}x=-{frac {1}{3}}left(a^{2}-x^{ 2}Nacht)^{frac {3}{2}} C}$

${displaystyle int x^{2}{sqrt {a^{2}-x^{2}}},{m {d}}x={frac {x}{8}}links(2x^{2}- a^{2}ight){sqrt {a^{2}-x^{2}}} {frac {a^{4}}{8}}arcsin {frac {x}{a}} C,(a >0)}$

${displaystyle int {frac {1}{x{sqrt {a^{2}-x^{2}}}}},{m {d}}x=-{frac {1}{a}}ln left| {frac {a {sqrt {a^{2}-x^{2}}}}{x}}ight| C}$

${displaystyle int {frac {x}{sqrt {a^{2}-x^{2}}}},{m {d}}x=-{sqrt {a^{2}-x^{2}} } C}$

${displaystyle int {frac {x^{2}}{sqrt {a^{2}-x^{2}}}},{m {d}}x=-{frac {x}{2}}{sqrt {a^{2}-x^{2}}} {frac {a^{2}}{2}}arcsin {frac {x}{a}} C,(a>0)}$

${displaystyle int {frac {sqrt {x^{2} a^{2}}}{x}},{m {d}}x={sqrt {x^{2} a^{2}}}-aln links|{frac {a {sqrt {x^{2} a^{2}}}}{x}}ight| C}$

${displaystyle int {frac {sqrt {x^{2}-a^{2}}}{x}},{m {d}}x={sqrt {x^{2}-a^{2}}} -aarccos {frac {a}{|x|}} C,(a>0)}$

${displaystyle int {frac {x^{2}}{sqrt {x^{2} a^{2}}}},{m {d}}x={frac {x{sqrt {x^{2} a ^{2}}}}{2}}-{frac {a^{2}}{2}}ln left(x {sqrt {x^{2} a^{2}}}ight) C}$

${displaystyle int {frac {1}{x{sqrt {x^{2} a^{2}}}}},{m {d}}x={frac {1}{a}}ln left|{frac {x}{a {sqrt {x^{2} a^{2}}}}}ight| C}$

${displaystyle int {frac {1}{x^{2}{sqrt {x^{2}pm a^{2}}}}},{m {d}}x=mp {frac {sqrt {x^{ 14}Uhr a^{2}}}{a^{2}x}} C}$

${displaystyle int {frac {1}{sqrt {x^{2}pm a^{2}}}},{m {d}}x=ln left|{frac {x {sqrt {x^{2}pm a^{2}}}}{a}}Nacht| C=Betreibername {arcsinh} {frac {x}{a}} C}$

${displaystyle int {frac {1}{sqrt {ax^{2} bx c}}},{m {d}}x={frac {1}{sqrt {a}}}ln left|2ax b 2{sqrt {a}}{Quadrat {ax^{2} bx c}}ight| C,(a>0)}$

${displaystyle int {frac {1}{sqrt {ax^{2} bx c}}},{m {d}}x={frac {1}{sqrt {-a}}}arcsin {frac {-2ax- b}{sqrt {b^{2}-4ac}}} C,(a<0)}$

${displaystyle int {sqrt {ax^{2} bx c}},{m {d}}x={frac {2ax b}{4a}}{sqrt {ax^{2} bx c}} {frac {4ac -b^{2}}{8a}}int {frac {1}{sqrt {ax^{2} bx c}}},{m {d}}x}$

${displaystyle int {frac {x}{sqrt {ax^{2} bx c}}},{m {d}}x={frac {sqrt {ax^{2} bx c}}{a}}-{ frac {b}{2a}}int {frac {1}{sqrt {ax^{2} bx c}}},{m {d}}x}$

${displaystyle int {frac {1}{x{sqrt {ax^{2} bx c}}}},{m {d}}x={frac {-1}{sqrt {c}}}ln left|{ frac {2{sqrt {c}}{sqrt {ax^{2} bx c}} bx 2c}{x}}ight| C,(c>0)}$

${displaystyle int {frac {1}{x{sqrt {ax^{2} bx c}}}},{m {d}}x={frac {1}{sqrt {-c}}}arcsin {frac { bx 2c}{|x|{sqrt {b^{2}-4ac}}}} C,(c<0)}$

${displaystyle int x^{3}{sqrt {x^{2} a^{2}}},{m {d}}x=left({frac {1}{5}}x^{2}-{ frac {2}{15}}a^{2}ight){sqrt {left(x^{2} a^{2}ight)^{3}}} C}$

${displaystyle int {frac {sqrt {x^{2}pm a^{2}}}{x^{4}}},{m {d}}x={frac {mp {sqrt {left(x^{ 2} a^{2}ight)^{3}}}}{3a^{2}x^{3}}} C}$

Exponentialfunktionen

${displaystyle int e^{x},{m {d}}x=e^{x} C}$

${displaystyle int a^{x},{m {d}}x={frac {a^{x}}{ln {a}}} C}$

${displaystyle int e^{ax},{m {d}}x={frac {e^{ax}}{a}} C}$

${displaystyle int a^{bx},{m {d}}x={frac {a^{bx}}{bln {a}}} C}$

${displaystyle int x^{n}e^{ax},{m {d}}x={frac {x^{n}e^{ax}}{a}}-{frac {n}{a}} int x^{n-1}e^{ax},{m {d}}x}$

Logarithmen

${displaystyle int ln {x},{m {d}}x=xln {x}-x C}$

${displaystyle int log _{b}{x},{m {d}}x=xlog _{b}{x}-xlog _{b}{e} C}$

${displaystyle int x^{n}ln ax,{m {d}}x=x^{n 1}left({frac {ln ax}{n 1}}-{frac {1}{(n 1)^ {2}}}Nacht) C}$

${displaystyle int x^{n}left(ln axight)^{m},{m {d}}x={frac {x^{n 1}}{n 1}}left(ln axight)^{m} -{frac {m}{n 1}}int x^{n}left(ln axight)^{m-1},{m {d}}x}$

Trigonometrische Funktionen

${displaystyle int sin {x},{m {d}}x=-cos {x} C}$

${displaystyle int cos {x},{m {d}}x=sin {x} C}$

${displaystyle int an {x},{m {d}}x=-ln {left|cos {x}ight|} C}$

${displaystyle int cot {x},{m {d}}x=ln {left|sin {x}ight|} C}$

${displaystyle int sec {x},{m {d}}x=ln {left|sec {x} an {x}ight|} C}$

${displaystyle int csc {x},{m {d}}x=ln {left|csc {x}-cot {x}ight|} C}$

${displaystyle int {frac {1}{sin x}},{m {d}}x=ln links| ein { frac {1}{2}}xight| C=ln links|{frac {1}{sin x}}-cot xight| C}$

${displaystyle int {frac {1}{cos x}},{m {d}}x=ln links| ein { frac {1}{2}}x { frac {1}{4}}pi ight| C=ln links|{frac {1}{cos x}} an xight| C}$

${displaystyle int arcsin {frac {x}{a}},{m {d}}x=xarcsin {frac {x}{a}} {sqrt {a^{2}-x^{2}}} C, (a>0)}$

${displaystyle int arccos {frac {x}{a}},{m {d}}x=xarccos {frac {x}{a}}-{sqrt {a^{2}-x^{2}}} C ,(a>0)}$

${displaystyle int arctan {frac {x}{a}},{m {d}}x=xarctan {frac {x}{a}}-{frac {a}{2}}ln left(a^{2} x^{2}ight) C,(a>0)}$

${displaystyle int {frac {1}{cos ^{2}x}},{m {d}}x=int sec ^{2}x,{m {d}}x= an x C}$

${displaystyle int {frac {1}{sin ^{2}x}},{m {d}}x=int csc ^{2}x,{m {d}}x=-cot x C}$

${displaystyle int sec {x}, an {x},{m {d}}x=sec {x} C}$

${displaystyle int csc {x},cot {x},{m {d}}x=-csc {x} C}$

${displaystyle int sin ^{2}x,{m {d}}x={ frac {1}{2}}(x-sin xcos x) C}$

${displaystyle int cos ^{2}x,{m {d}}x={ frac {1}{2}}(x sin xcos x) C}$

${displaystyle int sin ^{n}x,{m {d}}x=-{frac {sin ^{n-1}{x}cos {x}}{n}} {frac {n-1}{n }}int sin ^{n-2}{x},{m {d}}x}$

${displaystyle int cos ^{n}x,{m {d}}x={frac {cos ^{n-1}{x}sin {x}}{n}} {frac {n-1}{n} }int cos ^{n-2}{x},{m {d}}x}$

${displaystyle int an ^{n}x,{m {d}}x={frac { an ^{n-1}x}{n-1}}-int an ^{n-2}x,{m { d}}x,(neq 1)}$

${displaystyle int Kinderbett ^{n}x,{m {d}}x=-{frac {Kinderbett ^{n-1}x}{n-1}}-int Kinderbett ^{n-2}x,{m {d}}x,(neq 1)}$

${displaystyle int sec ^{n}x,{m {d}}x={frac { an xsec ^{n-2}x}{n-1}} {frac {n-2}{n-1}} int sek ^{n-2}x,{m {d}}x,(neq 1)}$

${displaystyle int csc ^{n}x,{m {d}}x=-{frac {cot xcsc ^{n-2}x}{n-1}} {frac {n-2}{n-1} }int csc ^{n-2}x,{m {d}}x,(neq 1)}$

${displaystyle int sin axsin bx,{m {d}}x={frac {sin(ab)x}{2(ab)}}-{frac {sin(ab)x}{2(ab)}} C, (a^{2}eq b^{2})}$

${displaystyle int sin axcos bx,{m {d}}x=-{frac {cos(ab)x}{2(ab)}}-{frac {cos(ab)x}{2(ab)}} C ,(a^{2}eq b^{2})}$

${displaystyle int cos axcos bx,{m {d}}x={frac {sin(ab)x}{2(ab)}} {frac {sin(ab)x}{2(ab)}} C,( a^{2}eq b^{2})}$

${displaystyle int sec x an x,{m {d}}x=sec x C}$

${displaystyle int csc xcot x,{m {d}}x=-csc x C}$

${displaystyle int cos ^{m}xsin ^{n}x,{m {d}}x={frac {cos ^{m-1}xsin ^{n 1}x}{mn}} {frac {m- 1}{mn}}int cos ^{m-2}xsin ^{n}x,{m {d}}x}$

${displaystyle =-{frac {sin ^{n-1}xcos ^{m 1}x}{mn}} {frac {n-1}{mn}}int cos ^{m}xsin ^{n-2} x,{m{d}}x}$

${displaystyle int x^{n}sin ax,{m {d}}x=-{frac {1}{a}}x^{n}cos ax {frac {n}{a}}int x^{n -1}cos ax,{m {d}}x}$

${displaystyle int x^{n}cos ax,{m {d}}x={frac {1}{a}}x^{n}sin ax-{frac {n}{a}}int x^{n -1}Sündenaxt,{m {d}}x}$

${displaystyle int e^{ax}sin bx,{m {d}}x={frac {e^{ax}left(asin bx-bcos bxight)}{a^{2} b^{2}}} C }$

${displaystyle int e^{ax}cos bx,{m {d}}x={frac {e^{ax}left(bsin bx acos bxight)}{a^{2} b^{2}}} C}$

Hyperbolische Funktionen

${displaystyle int sinh x,{m {d}}x=cosh x C}$

${displaystyle int cosh x,{m {d}}x=sinh x C}$

${displaystyle int anh x,{m {d}}x=ln |cosh x| C}$

${displaystyle int Operatorname {csch} ,x,{m {d}}x=ln left| anh {x über 2}Nacht| C}$

${displaystyle int Operatorname {sech} ,x,{m {d}}x=arctan(sinh x) C}$

${displaystyle int coth x,{m {d}}x=ln |sinh x| C}$

${displaystyle int sinh ^{2}x,{m {d}}x={frac {1}{4}}sinh 2x-{frac {1}{2}}x C}$

${displaystyle int cosh ^{2}x,{m {d}}x={frac {1}{4}}sinh 2x {frac {1}{2}}x C}$

${displaystyle int Operatorname {sech} ^{2}x,{m {d}}x= anh x C}$

${displaystyle int {frac {1}{sinh ^{2}x}},{m {d}}x= Operatorname {coth } x C}$

${displaystyle int sinh ^{-1}{frac {x}{a}},{m {d}}x=xsinh ^{-1}{frac {x}{a}}-{sqrt {x^{2 } a^{2}}} C}$

${displaystyle int cosh ^{-1}{frac {x}{a}},{m {d}}x=xcosh ^{-1}{frac {x}{a}}-{sqrt {x^{2 }-a^{2}}} Cleft(cosh ^{-1}{frac {x}{a}}>0,a>0ight)}$

${displaystyle int cosh ^{-1}{frac {x}{a}},{m {d}}x=xcosh ^{-1}{frac {x}{a}} {sqrt {x^{2} -a^{2}}} Cleft(cosh ^{-1}{frac {x}{a}}<0,a>0ight)}$

${displaystyle int anh ^{-1}{frac {x}{a}},{m {d}}x=x anh ^{-1}{frac {x}{a}} {frac {a}{2 }}ln links|a^{2}-x^{2}ight| C}$

${displaystyle int Operatorname {sech} x anh x,{m {d}}x= Operatorname {sech} x C}$

${displaystyle int Operatorname {csch} xcoth x,{m {d}}x= Operatorname {csch} x C}$