integral from 1 to e of x^{1/2}ln(x)

Lösung

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

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

Dezimale

1.44037 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

3 \cdot \frac{4}{9} x^{\frac{3}{2}} = \frac{4}{3} x^{\frac{3}{2}}

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

Berechne die Grenzen:

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

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

Vereinfache

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

\int_{0}^{π} arctan (x) d x

\int_{- 1}^{2} (x^{3} - 3 x^{2} - x + 3) d x

\int_{0}^{1} 6 π x^{2} d x

\int_{0}^{\frac{1}{2}} 2 x + 3 - x e^{3 x} d x

\int_{- 4}^{2} x^{2} + 4 x d x