integral from 1/2 to 2 of |ln(x)|

Lösung

\int_{\frac{1}{2}}^{2} ∣ ln (x) ∣ d x

- \frac{1}{2} + \frac{3}{2} ln (2)

Dezimale

0.53972 \dots

Schritte zur Lösung

\int_{\frac{1}{2}}^{2} ∣ ln (x) ∣ d x

Wende U-Substitution an

= \int_{- l n (2)}^{l n (2)} ∣ u ∣ e^{u} d u

Entferne die Extremwerte

= \int_{- l n (2)}^{0} (- u) e^{u} d u + \int_{0}^{l n (2)} u e^{u} d u

\int_{- l n (2)}^{0} (- u) e^{u} d u = \frac{1}{2} - \frac{1}{2} ln (2)

\int_{0}^{l n (2)} u e^{u} d u = 2 ln (2) - 1

= \frac{1}{2} - \frac{1}{2} ln (2) + 2 ln (2) - 1

Vereinfache

\frac{1}{2} - \frac{1}{2} ln (2) + 2 ln (2) - 1 : - \frac{1}{2} + \frac{3}{2} ln (2)

= - \frac{1}{2} + \frac{3}{2} ln (2)

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