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Integration durch Substitution
i-08-01
expression
144
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Bestimmen Sie \displaystyle \int f[5] dx.

Verwenden Sie C als Integrationskonstante.

f[4] + C

Bei der Integration durch Substitution verwenden wir die Gleichung

\displaystyle \int f\left(g(x)\right)g'(x) dx = F(g(x)) + C,

wobei F eine Stammfunktion von f ist.

Es geht also darum, Funktionen f und g so zu identifizieren, dass f\left(g(x)\right)g'(x) gleich dem Integranden f[5] ist und wir eine Stammfunktion F von f bestimmen können.

Hier eignen sich f(x) = f[1] und g(x) = f[2] mit g'(x) = f[3].

Es sind dann F(x) = f[0] eine Stammfunktion von f und F(g(x)) = f[4].

Damit ergibt sich

\displaystyle \int f[5] dx = f[4] + C.