integral from 0 to 2 of (30t^2)/(4+t^3)

Lösung

\int_{0}^{2} \frac{30 t ^{2}}{4 + t ^{3}} d t

10 ln (3)

Dezimale

10.98612 \dots

Schritte zur Lösung

\int_{0}^{2} \frac{30 t ^{2}}{4 + t ^{3}} d t

Entferne die Konstante:

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

= 30 \cdot \int_{0}^{2} \frac{t ^{2}}{4 + t ^{3}} d t

Wende U-Substitution an

= 30 \cdot \int_{4}^{12} \frac{1}{3 u} d u

Entferne die Konstante:

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

= 30 \cdot \frac{1}{3} \cdot \int_{4}^{12} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 30 \cdot \frac{1}{3} [ln ∣ u ∣]_{4}^{12}

Vereinfache

30 \cdot \frac{1}{3} [ln ∣ u ∣]_{4}^{12} : 10 [ln ∣ u ∣]_{4}^{12}

= 10 [ln ∣ u ∣]_{4}^{12}

Berechne die Grenzen:

ln (12) - 2 ln (2)

= 10 (ln (12) - 2 ln (2))

Vereinfache

= 10 ln (3)

\int \frac{x ^{3} + x}{x + 1} d x

\int sin (5) (t) cos (4) (t) d t

x \to \infty lim (6 x^{3})

d er i v a t i v e 2 π r - π r^{2}

n = 0 \sum \infty (\frac{2 x}{3})^{n}