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Beliebt Rechnen >

integral from 1 to e of x^2ln(sqrt(x))

Lösung

\int_{1}^{e} x^{2} ln (x) d x

Lösung

\frac{2 e ^{3} + 1}{18}

+1

Dezimale

2.28728 \dots

Schritte zur Lösung

\int_{1}^{e} x^{2} ln (x) d x

Vereinfache

ln (x) : \frac{1}{2} ln (x)

= \int_{1}^{e} \frac{1}{2} x^{2} ln (x) d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{1}^{e} x^{2} ln (x) d x

Wende die partielle Integration an

= \frac{1}{2} [\frac{1}{3} (x^{3} ln (x) - 3 \cdot \int \frac{x ^{2}}{3} d x)]_{1}^{e}

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= \frac{1}{2} [\frac{1}{3} (x^{3} ln (x) - 3 \cdot \frac{x ^{3}}{9})]_{1}^{e}

3 \cdot \frac{x ^{3}}{9} = \frac{x ^{3}}{3}

= \frac{1}{2} [\frac{1}{3} (x^{3} ln (x) - \frac{x ^{3}}{3})]_{1}^{e}

Berechne die Grenzen:

\frac{2 e ^{3}}{9} + \frac{1}{9}

= \frac{1}{2} (\frac{2 e ^{3}}{9} + \frac{1}{9})

Vereinfache

= \frac{2 e ^{3} + 1}{18}

Graph

Beliebte Beispiele

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d er i v a t i v e e^{x^{2} + 9 x + 7}

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\int 9 x^{\frac{2}{3}} d x