integral of x(3x+5)^2

Lösung

\int x (3 x + 5)^{2} d x

\frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C

Schritte zur Lösung

\int x (3 x + 5)^{2} d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} (3 x + 5)^{2} - \int 3 x^{2} (3 x + 5) d x

\int 3 x^{2} (3 x + 5) d x = \frac{9 x ^{4}}{4} + 5 x^{3}

= \frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3})

Vereinfache

\frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3}) : \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C

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