integral of x*ln((1-x)/(1+x))

Lösung

\int x \cdot ln (\frac{1 - x}{1 + x}) d x

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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

\int_{1}^{2} \frac{x}{2} - \frac{2}{x} d x

x \to \frac{π}{3} lim (tan (3 x))

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

x \to 1 lim (3 x - 1)