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Beliebt Rechnen >

integral from 1 to e of 4x^2ln(x)

Lösung

\int_{1}^{e} 4 x^{2} ln (x) d x

Lösung

\frac{4 ( 2 e ^{3} + 1 )}{9}

+1

Dezimale

18.29825 \dots

Schritte zur Lösung

\int_{1}^{e} 4 x^{2} ln (x) d x

Entferne die Konstante:

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

= 4 \cdot \int_{1}^{e} x^{2} ln (x) d x

Wende die partielle Integration an

= 4 [\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}

= 4 [\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}

= 4 [\frac{1}{3} (x^{3} ln (x) - \frac{x ^{3}}{3})]_{1}^{e}

Berechne die Grenzen:

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

= 4 (\frac{2 e ^{3}}{9} + \frac{1}{9})

Vereinfache

= \frac{4 ( 2 e ^{3} + 1 )}{9}

Graph

Beliebte Beispiele

derivative of 1/(sqrt(1+x))

\frac{d}{d x} (\frac{1}{1 + x})

derivative of e^{1/2}

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

integral of (1-x)/(x^3)

\int \frac{1 - x}{x ^{3}} d x

(dy)/(dx)=(x-18)^2

\frac{d y}{d x} = (x - 18)^{2}

limit as x approaches 4 of 9/x

x \to 4 lim (\frac{9}{x})