integral from 0 to 2 of (6x^2)/(2+x^3)

Lösung

\int_{0}^{2} \frac{6 x ^{2}}{2 + x ^{3}} d x

2 ln (5)

Dezimale

3.21887 \dots

Schritte zur Lösung

\int_{0}^{2} \frac{6 x ^{2}}{2 + x ^{3}} d x

Entferne die Konstante:

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

= 6 \cdot \int_{0}^{2} \frac{x ^{2}}{2 + x ^{3}} d x

Wende U-Substitution an

= 6 \cdot \int_{2}^{10} \frac{1}{3 u} d u

Entferne die Konstante:

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

= 6 \cdot \frac{1}{3} \cdot \int_{2}^{10} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 6 \cdot \frac{1}{3} [ln ∣ u ∣]_{2}^{10}

Vereinfache

6 \cdot \frac{1}{3} [ln ∣ u ∣]_{2}^{10} : 2 [ln ∣ u ∣]_{2}^{10}

= 2 [ln ∣ u ∣]_{2}^{10}

Berechne die Grenzen:

ln (10) - ln (2)

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

Vereinfache

= 2 ln (5)

\int_{1}^{5} ln (2 x) - ln (x) d x

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

\int_{0}^{2 π} 2 cos^{2} (t) d t

\int_{1}^{4} \frac{ln ( x )}{x} d x

\int_{0}^{1} e^{1 - x} d x