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

Lösung

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

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

Dezimale

4.57456 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

Berechne die Grenzen:

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

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

Vereinfache

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

x \to 0 lim (\frac{3 x}{tan ( x )})

y^{^{'}} - 2 x y = 2 x

x \to 1 lim (arctan (x))

y^{^{'}} - \frac{4}{x} y = \frac{1}{x ^{3}} y^{4}

\frac{d}{d x} (- 30 sin (x))