integral from 1 to 8 of (12x)/(x^2+6)

Lösung

\int_{1}^{8} \frac{12 x}{x ^{2} + 6} d x

6 ln (10)

Dezimale

13.81551 \dots

Schritte zur Lösung

\int_{1}^{8} \frac{12 x}{x ^{2} + 6} d x

Entferne die Konstante:

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

= 12 \cdot \int_{1}^{8} \frac{x}{x ^{2} + 6} d x

Wende U-Substitution an

= 12 \cdot \int_{7}^{70} \frac{1}{2 u} d u

Entferne die Konstante:

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

= 12 \cdot \frac{1}{2} \cdot \int_{7}^{70} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 12 \cdot \frac{1}{2} [ln ∣ u ∣]_{7}^{70}

Vereinfache

12 \cdot \frac{1}{2} [ln ∣ u ∣]_{7}^{70} : 6 [ln ∣ u ∣]_{7}^{70}

= 6 [ln ∣ u ∣]_{7}^{70}

Berechne die Grenzen:

ln (70) - ln (7)

= 6 (ln (70) - ln (7))

Vereinfache

= 6 ln (10)

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