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Beliebt Rechnen >

integral of atln(t+b)

Lösung

\int a t ln (t + b) d t

Lösung

a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})) + C

Schritte zur Lösung

\int a t ln (t + b) d t

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a \cdot \int t ln (t + b) d t

Wende die partielle Integration an

= a (\frac{1}{2} t^{2} ln (t + b) - \int \frac{t ^{2}}{2 ( t + b )} d t)

\int \frac{t ^{2}}{2 ( t + b )} d t = \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})

= a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2}))

Füge eine Konstante zur Lösung hinzu

= a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})) + C

Graph

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