integral of (3x^2+2)ln(2x)

Lösung

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

x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x + C

Schritte zur Lösung

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

Wende die partielle Integration an

= ln (2 x) (x^{3} + 2 x) - \int x^{2} + 2 d x

\int x^{2} + 2 d x = \frac{x ^{3}}{3} + 2 x

= ln (2 x) (x^{3} + 2 x) - (\frac{x ^{3}}{3} + 2 x)

Vereinfache

ln (2 x) (x^{3} + 2 x) - (\frac{x ^{3}}{3} + 2 x) : x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x

= x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x

Füge eine Konstante zur Lösung hinzu

= x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x + C

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

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