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Beliebt Rechnen >

integral of-xln(2x^2+1)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

\int \frac{2 x ^{3}}{2 x ^{2} + 1} d x = 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1)

= - (\frac{1}{2} x^{2} ln (2 x^{2} + 1) - 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1))

Vereinfache

- (\frac{1}{2} x^{2} ln (2 x^{2} + 1) - 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1)) : - \frac{1}{2} x^{2} ln (2 x^{2} + 1) + \frac{x ^{2}}{2} - \frac{1}{4} ln 2 x^{2} + 1

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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