integral from 2 to 7 of (10x)/(x^2+1)

Lösung

\int_{2}^{7} \frac{10 x}{x ^{2} + 1} d x

5 ln (10)

Dezimale

11.51292 \dots

Schritte zur Lösung

\int_{2}^{7} \frac{10 x}{x ^{2} + 1} d x

Entferne die Konstante:

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

= 10 \cdot \int_{2}^{7} \frac{x}{x ^{2} + 1} d x

Wende U-Substitution an

= 10 \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

= 10 \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 ∣)

= 10 \cdot \frac{1}{2} [ln ∣ u ∣]_{5}^{50}

Vereinfache

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

= 5 [ln ∣ u ∣]_{5}^{50}

Berechne die Grenzen:

ln (50) - ln (5)

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

Vereinfache

= 5 ln (10)

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

\int_{1}^{2} (3 x^{2} + 4 x^{3}) d x

\int_{0}^{1} - y ln (y) d y

\int_{3}^{7} 15 ln (x + 5) d x

\int_{1}^{2} 2 x - 2 d x