integral from 0 to 16 of (500x)/(x^2+24)

Lösung

\int_{0}^{16} \frac{500 x}{x ^{2} + 24} d x

250 ln (\frac{35}{3})

Dezimale

614.18394 \dots

Schritte zur Lösung

\int_{0}^{16} \frac{500 x}{x ^{2} + 24} d x

Entferne die Konstante:

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

= 500 \cdot \int_{0}^{16} \frac{x}{x ^{2} + 24} d x

Wende U-Substitution an

= 500 \cdot \int_{24}^{280} \frac{1}{2 u} d u

Entferne die Konstante:

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

= 500 \cdot \frac{1}{2} \cdot \int_{24}^{280} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= 500 \cdot \frac{1}{2} [ln ∣ u ∣]_{24}^{280}

Vereinfache

500 \cdot \frac{1}{2} [ln ∣ u ∣]_{24}^{280} : 250 [ln ∣ u ∣]_{24}^{280}

= 250 [ln ∣ u ∣]_{24}^{280}

Berechne die Grenzen:

ln (280) - ln (24)

= 250 (ln (280) - ln (24))

Vereinfache

= 250 ln (\frac{35}{3})

d er i v a t i v e 5 π x^{2}

\frac{\partial}{\partial x} (40 x^{\frac{3}{4}} y^{\frac{1}{4}})

a re a y = x^{3}, x = - 1

t an g e n t f (x) = (x^{3} + 1) (x^{2} - 2), at x = 2

t an g e n t f (x) = (x - 4) (3 - x)^{3}, at x = 5