integral of x^3ln(1+x^2)

Lösung

\int x^{3} ln (1 + x^{2}) d x

\frac{1}{4} x^{4} ln (1 + x^{2}) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 1) + C

Schritte zur Lösung

\int x^{3} ln (1 + x^{2}) d x

Wende die partielle Integration an

= \frac{1}{4} x^{4} ln (1 + x^{2}) - \int \frac{x ^{5}}{2 ( x ^{2} + 1 )} d x

\int \frac{x ^{5}}{2 ( x ^{2} + 1 )} d x = \frac{1}{2} (\frac{x ^{4}}{4} - \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 1)

= \frac{1}{4} x^{4} ln (1 + x^{2}) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} ln (1 + x^{2}) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 1) + C