integral from 1 to e of 4/(x(10+9ln(x)))

Lösung

\int_{1}^{e} \frac{4}{x ( 10 + 9 ln ( x ) )} d x

\frac{4}{9} (ln (19) - ln (10))

Dezimale

0.28526 \dots

Schritte zur Lösung

\int_{1}^{e} \frac{4}{x ( 10 + 9 ln ( x ) )} d x

Entferne die Konstante:

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

= 4 \cdot \int_{1}^{e} \frac{1}{x ( 10 + 9 ln ( x ) )} d x

Wende U-Substitution an

= 4 \cdot \int_{10}^{19} \frac{1}{9 u} d u

Entferne die Konstante:

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

= 4 \cdot \frac{1}{9} \cdot \int_{10}^{19} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{9} [ln ∣ u ∣]_{10}^{19}

Vereinfache

4 \cdot \frac{1}{9} [ln ∣ u ∣]_{10}^{19} : \frac{4}{9} [ln ∣ u ∣]_{10}^{19}

= \frac{4}{9} [ln ∣ u ∣]_{10}^{19}

Berechne die Grenzen:

ln (19) - ln (10)

= \frac{4}{9} (ln (19) - ln (10))

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