integral of (1-x)ln(x)

Lösung

\int (1 - x) ln (x) d x

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

Schritte zur Lösung

\int (1 - x) ln (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{sin ( 2 x )}{1 + sin ^{2} ( x )} d x

\frac{\partial}{\partial y} (sin (x^{3} y^{2}))

\frac{\partial}{\partial y} (3 x^{2} y^{4} - 4)

d er i v a t i v e \frac{2}{x ^{2} + 3}

d er i v a t i v e 2 a x + b