integral from 0 to 1 of (4x)/(1+x^2)

Lösung

\int_{0}^{1} \frac{4 x}{1 + x ^{2}} d x

2 ln (2)

Dezimale

1.38629 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{4 x}{1 + x ^{2}} d x

Entferne die Konstante:

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

= 4 \cdot \int_{0}^{1} \frac{x}{1 + x ^{2}} d x

Wende U-Substitution an

= 4 \cdot \int_{1}^{2} \frac{1}{2 u} d u

Entferne die Konstante:

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

= 4 \cdot \frac{1}{2} \cdot \int_{1}^{2} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{2} [ln ∣ u ∣]_{1}^{2}

Vereinfache

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

= 2 [ln ∣ u ∣]_{1}^{2}

Berechne die Grenzen:

ln (2)

= 2 ln (2)

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

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