integral of 3x(3x+4)^4

Lösung

\int 3 x (3 x + 4)^{4} d x

3 (\frac{1}{2} x^{2} (3 x + 4)^{4} - 6 (\frac{9 x ^{6}}{2} + \frac{108 x ^{5}}{5} + 36 x^{4} + \frac{64 x ^{3}}{3})) + C

Schritte zur Lösung

\int 3 x (3 x + 4)^{4} d x

Entferne die Konstante:

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

= 3 \cdot \int x (3 x + 4)^{4} d x

Wende die partielle Integration an

= 3 (\frac{1}{2} x^{2} (3 x + 4)^{4} - \int 6 x^{2} (3 x + 4)^{3} d x)

\int 6 x^{2} (3 x + 4)^{3} d x = 6 (\frac{9 x ^{6}}{2} + \frac{108 x ^{5}}{5} + 36 x^{4} + \frac{64 x ^{3}}{3})

= 3 (\frac{1}{2} x^{2} (3 x + 4)^{4} - 6 (\frac{9 x ^{6}}{2} + \frac{108 x ^{5}}{5} + 36 x^{4} + \frac{64 x ^{3}}{3}))

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{1}{2} x^{2} (3 x + 4)^{4} - 6 (\frac{9 x ^{6}}{2} + \frac{108 x ^{5}}{5} + 36 x^{4} + \frac{64 x ^{3}}{3})) + C

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