integral from 2 to 3 of (2x)/((1+x^2))

Lösung

\int_{2}^{3} \frac{2 x}{( 1 + x ^{2} )} d x

ln (2)

Dezimale

0.69314 \dots

Schritte zur Lösung

\int_{2}^{3} \frac{2 x}{1 + x ^{2}} d x

Entferne die Konstante:

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

= 2 \cdot \int_{2}^{3} \frac{x}{1 + x ^{2}} d x

Wende U-Substitution an

= 2 \cdot \int_{5}^{10} \frac{1}{2 u} d u

Entferne die Konstante:

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

= 2 \cdot \frac{1}{2} \cdot \int_{5}^{10} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} [ln ∣ u ∣]_{5}^{10}

Vereinfache

2 \cdot \frac{1}{2} [ln ∣ u ∣]_{5}^{10} : [ln ∣ u ∣]_{5}^{10}

= [ln ∣ u ∣]_{5}^{10}

Berechne die Grenzen:

ln (10) - ln (5)

= ln (10) - ln (5)

Vereinfache

= ln (2)

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