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

Lösung

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

\frac{4 e ^{\frac{3}{2}} + 8}{9}

Dezimale

2.88075 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

= [\frac{1}{3} (2 x^{\frac{3}{2}} ln (x^{2}) - 3 \cdot \int \frac{4 x ^{\frac{1}{2}}}{3} d x)]_{1}^{e}

\int \frac{4 x ^{\frac{1}{2}}}{3} d x = \frac{8}{9} x^{\frac{3}{2}}

= [\frac{1}{3} (2 x^{\frac{3}{2}} ln (x^{2}) - 3 \cdot \frac{8}{9} x^{\frac{3}{2}})]_{1}^{e}

Vereinfache

[\frac{1}{3} (2 x^{\frac{3}{2}} ln (x^{2}) - 3 \cdot \frac{8}{9} x^{\frac{3}{2}})]_{1}^{e} : [\frac{1}{3} (4 x^{\frac{3}{2}} ln (x) - \frac{8}{3} x^{\frac{3}{2}})]_{1}^{e}

= [\frac{1}{3} (4 x^{\frac{3}{2}} ln (x) - \frac{8}{3} x^{\frac{3}{2}})]_{1}^{e}

Berechne die Grenzen:

\frac{4 e ^{\frac{3}{2}}}{9} + \frac{8}{9}

= \frac{4 e ^{\frac{3}{2}}}{9} + \frac{8}{9}

Vereinfache

= \frac{4 e ^{\frac{3}{2}} + 8}{9}

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