Seien a und b fest gew\344hlte reelle Zahlen. Das (bestimmte) Integral einer Funktion f von a bis b l\344sst durch die Funktion "int(f(x), x=a..b)" (bzw. "value(Int(f(x), x=a..b))") berechnen. Beispiel:Int(x^2, x=-1..2); value(Int(x^2, x=-1..2)); int(x^2, x=-1..2);

NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkKiRJInhHRigiIiMvRis7ISIiRiw=

NiMiIiQ=NiMiIiQ=

F\374r eine Funktion l\344sst sich allerdings auch schlichtweg die Stammfunktion angeben, ohne dass die Grenzen ber\374cksichtigt werden. Beispiel:Int(x^2, x); value(Int(x^2, x)); int(x^2, x);

NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkKiRJInhHRigiIiNGKw==

NiMsJCokSSJ4RzYiIiIkIyIiIkYnNiMsJCokSSJ4RzYiIiIkIyIiIkYn

Bei unbestimmten Integralen ist mindestens eine der Grenzen "variabel". Auch derartige Berechnungen sind in Maple m\366glich. Beispiele:int(x^2, x=a..2); (Beispiel mit einer variablen unteren Grenze) int(x^2, x=-1..b); (Beispiel mit einer variablen oberen Grenze) int(x^2, x=a..b); (Beispiel mit zwei variablen Grenzen)

NiMsJiMiIikiIiQiIiIqJEkiYUc2IkYmIyEiIkYmNiMsJiokSSJiRzYiIiIkIyIiIkYnRihGKQ==NiMsJiokSSJiRzYiIiIkIyIiIkYnKiRJImFHRiZGJyMhIiJGJw==

Es ist nat\374rlich durchaus m\366glich mindestens eine der Grenzen a=-infinity und/oder b=infinity, also "unendlich", zu w\344hlen. Besitzt das Integral nach der Berechnung einen endlichen Wert, so bezeichnet man ein solches Integral als uneigentliches Integral. Diese k\366nnen ebenfalls mit Maple berechnet werden. Beispiel:int(1/(1+x^2), x=-1..infinity); int(1/(1+x^2), x=-infinity..2); int(1/(1+x^2), x=-infinity..infinity);

NiMsJEkjUGlHSSpwcm90ZWN0ZWRHRiUjIiIkIiIlNiMsJkkjUGlHSSpwcm90ZWN0ZWRHRiUjIiIiIiIjLUknYXJjdGFuRzYkRiVJKF9zeXNsaWJHNiI2I0YoRic=

NiNJI1BpR0kqcHJvdGVjdGVkR0Yk

Weitere Beispiele sindint(sin(x)/x, x=0..infinity); int(sin(x^2), x=0..infinity);

NiMsJEkjUGlHSSpwcm90ZWN0ZWRHRiUjIiIiIiIjNiMsJComIiIjIyIiIkYlSSNQaUdJKnByb3RlY3RlZEdGKUYmI0YnIiIl

Nicht zu den uneigentlichen Integralen geh\366rt beispielsweise:int(x^2, x=-1..infinity);

NiNJKWluZmluaXR5R0kqcHJvdGVjdGVkR0Yk

Stammfunktionen von Polynomen:int(x^s, x) assuming s<>-1; int(1/x, x); int(1/sqrt(1-x^2), x) assuming abs(x)<1; int(1/(1+x^2), x); int(1/(1-x^2), x); int(1/sqrt(1+x^2), x); int(sqrt(1-x^2), x); int(1/sqrt(x^2-1), x); int(1/(a*x^2+b*x+c), x) assuming a::real, b::real, c::real; int(1/(1+x^4), x);

NiMqJilJInhHNiIsJiIiIkYoSSNzfGlyR0YmRihGKEYnISIiNiMtSSNsbkc2JEkqcHJvdGVjdGVkR0YmSShfc3lzbGliRzYiNiNJInhHRig=NiMtSSdhcmNzaW5HNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjSSJ4R0YoNiMtSSdhcmN0YW5HNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjSSJ4R0YoNiMtSShhcmN0YW5oRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kieEdGKA==

NiMtSShhcmNzaW5oRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kieEdGKA==NiMsJiomSSJ4RzYiIiIiLCZGJ0YnKiRGJSIiIyEiIiNGJ0YqRiwtSSdhcmNzaW5HNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJjYjRiVGLA==NiMtSSNsbkc2JEkqcHJvdGVjdGVkR0YmSShfc3lzbGliRzYiNiMsJkkieEdGKCIiIiokLCYhIiJGLCokRisiIiNGLCNGLEYxRiw=NiMsJComLCYqJkkiY0c2IiIiIkkiYUdGKEYpIiIlKiRJImJHRigiIiMhIiIjRi9GLi1JJ2FyY3Rhbkc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YoNiMqJiwmKiZGKkYpSSJ4R0YoRilGLkYtRilGKUYlRjBGKUYuNiMsKComIiIjIyIiIkYlLUkjbG5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkc2IjYjKiYsKCokSSJ4R0YtRiVGJyomRjJGJ0YlRiZGJ0YnRidGJywoRjFGJ0YzISIiRidGJ0Y1RicjRiciIikqJkYlRiYtSSdhcmN0YW5HRio2IywmRjNGJ0YnRidGJyNGJyIiJSomRiVGJi1GOjYjLCZGM0YnRjVGJ0YnRj0=

Stammfunktionen trigonometrischer Funktionen:int(sin(x), x); int(cos(x), x); int(tan(x), x); int(sinh(x), x); int(cosh(x), x); int(tanh(x), x); int(arcsin(x), x); int(arccos(x), x); int(arctan(x), x); int(arcsinh(x), x); int(arccosh(x), x); int(arctanh(x), x); int(sin(n*x)*sin(m*x), x) assuming m::posint, n::posint; int(sin(n*x)*cos(m*x), x) assuming m::posint, n::posint; int(cos(n*x)*cos(m*x), x) assuming m::posint, n::posint; int(1/(cos(x)^2), x);

NiMsJC1JJGNvc0c2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiNJInhHRikhIiI=NiMtSSRzaW5HNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjSSJ4R0YoNiMsJC1JI2xuRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2Iy1JJGNvc0dGJjYjSSJ4R0YpISIiNiMtSSVjb3NoRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kieEdGKA==NiMtSSVzaW5oRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kieEdGKA==NiMtSSNsbkc2JEkqcHJvdGVjdGVkR0YmSShfc3lzbGliRzYiNiMtSSVjb3NoR0YlNiNJInhHRig=NiMsJiomSSJ4RzYiIiIiLUknYXJjc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0YlRidGJyokLCZGJ0YnKiRGJSIiIyEiIiNGJ0YxRic=NiMsJiomSSJ4RzYiIiIiLUknYXJjY29zRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0YlRidGJyokLCZGJ0YnKiRGJSIiIyEiIiNGJ0YxRjI=NiMsJiomSSJ4RzYiIiIiLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0YlRidGJy1JI2xuR0YqNiMsJkYnRicqJEYlIiIjRicjISIiRjM=NiMsJiomSSJ4RzYiIiIiLUkoYXJjc2luaEc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YmNiNGJUYnRicqJCwmRidGJyokRiUiIiNGJyNGJ0YxISIiNiMsJiomSSJ4RzYiIiIiLUkoYXJjY29zaEc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YmNiNGJUYnRicqJiwmRiVGJyEiIkYnI0YnIiIjLCZGJUYnRidGJ0YxRjA=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

with(plots): f:= x -> 1/(1 - x + x^4); Graph := plot(f(x), x = -4..4, thickness = 2): Fl\344che := seq(plot ([[i/30, 0], [i/30, f(i/30)]], color = gray, thickness = 3), i = 1..30): TextA := textplot ([1/2, 0.9, `A`], font = [TIMES, BOLD, 25]): display([Fl\344che, Graph, TextA]);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsKCIiIkYuOSQhIiIqJEYvIiIlRi5GMEYlRiVGJQ==

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