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Beliebt Rechnen >

integral of xsqrt(ln(x))

Lösung

\int x ln (x) d x

Lösung

\frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x))) + C

Schritte zur Lösung

\int x ln (x) d x

Wende U-Substitution an

= \int e^{2 u} u d u

Wende U-Substitution an

= \int e^{v} \frac{v}{2} \frac{1}{2} d v

Entferne die Konstante:

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

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

Wende die partielle Integration an

= \frac{1}{2} (e^{v} \frac{v}{2} - \int \frac{e ^{v}}{2 2 v} d v)

\int \frac{e ^{v}}{2 2 v} d v = \frac{π}{2 2} erfi (v)

= \frac{1}{2} (e^{v} \frac{v}{2} - \frac{π}{2 2} erfi (v))

Ersetze zurück

= \frac{1}{2} (e^{2 l n (x)} \frac{2 ln ( x )}{2} - \frac{π}{2 2} erfi (2 ln (x)))

Vereinfache

\frac{1}{2} (e^{2 l n (x)} \frac{2 ln ( x )}{2} - \frac{π}{2 2} erfi (2 ln (x))) : \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x)))

= \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x))) + C

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