integral from 2 to 7 of 2/(8x-11)

Lösung

\int_{2}^{7} \frac{2}{8 x - 11} d x

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

Dezimale

0.54930 \dots

Schritte zur Lösung

\int_{2}^{7} \frac{2}{8 x - 11} d x

Entferne die Konstante:

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

= 2 \cdot \int_{2}^{7} \frac{1}{8 x - 11} d x

Wende U-Substitution an

= 2 \cdot \int_{5}^{45} \frac{1}{8 u} d u

Entferne die Konstante:

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

= 2 \cdot \frac{1}{8} \cdot \int_{5}^{45} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{8} [ln ∣ u ∣]_{5}^{45}

Vereinfache

2 \cdot \frac{1}{8} [ln ∣ u ∣]_{5}^{45} : \frac{1}{4} [ln ∣ u ∣]_{5}^{45}

= \frac{1}{4} [ln ∣ u ∣]_{5}^{45}

Berechne die Grenzen:

ln (45) - ln (5)

= \frac{1}{4} (ln (45) - ln (5))

Vereinfache

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

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