integral from 0 to 0.5 of 1/(1-x)

Lösung

\int_{0}^{0.5} \frac{1}{1 - x} d x

0.69314 \dots

Schritte zur Lösung

\int_{0}^{0.5} \frac{1}{1 - x} d x

Wende U-Substitution an

= \int_{1}^{0.5} - \frac{1}{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0.5}^{1} - \frac{1}{u} d u

Entferne die Konstante:

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

= - (- \int_{0.5}^{1} \frac{1}{u} d u)

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- [ln ∣ u ∣]_{0.5}^{1})

Vereinfache

= [ln ∣ u ∣]_{0.5}^{1}

Berechne die Grenzen:

0.69314 \dots

= 0.69314 \dots