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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{ln ( u )}{2 u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{ln ( u )}{u} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int v d v

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{v ^{2}}{2}

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int sec (a) x d x

\frac{\partial}{\partial x} (4 x^{3} + z^{2})

\frac{d}{d x} ((r (x) (x)^{2} - x^{2})^{\frac{1}{2}})

t an g e n t f (x) = sin (3 x), (\frac{π}{18}, \frac{2}{2})

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