integral from 2 to 7 of (8x)/(x^2+1)

Lösung

\int_{2}^{7} \frac{8 x}{x ^{2} + 1} d x

4 ln (10)

Dezimale

9.21034 \dots

Schritte zur Lösung

\int_{2}^{7} \frac{8 x}{x ^{2} + 1} d x

Entferne die Konstante:

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

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

Wende U-Substitution an

= 8 \cdot \int_{5}^{50} \frac{1}{2 u} d u

Entferne die Konstante:

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

= 8 \cdot \frac{1}{2} \cdot \int_{5}^{50} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{2} [ln ∣ u ∣]_{5}^{50}

Vereinfache

8 \cdot \frac{1}{2} [ln ∣ u ∣]_{5}^{50} : 4 [ln ∣ u ∣]_{5}^{50}

= 4 [ln ∣ u ∣]_{5}^{50}

Berechne die Grenzen:

ln (50) - ln (5)

= 4 (ln (50) - ln (5))

Vereinfache

= 4 ln (10)

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