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Beliebt Rechnen >

integral of (x+4)/(x^2+11x+18)

Lösung

\int \frac{x + 4}{x ^{2} + 11 x + 18} d x

Lösung

\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x + 9 ∣ + C

Schritte zur Lösung

\int \frac{x + 4}{x ^{2} + 11 x + 18} d x

Wende U-Substitution an

= \int \frac{u}{u ^{2} + 3 u - 10} d u

Ermittle den Partialbruch von

\frac{u}{u ^{2} + 3 u - 10} : \frac{2}{7 ( u - 2 )} + \frac{5}{7 ( u + 5 )}

= \int \frac{2}{7 ( u - 2 )} + \frac{5}{7 ( u + 5 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2}{7 ( u - 2 )} d u + \int \frac{5}{7 ( u + 5 )} d u

\int \frac{2}{7 ( u - 2 )} d u = \frac{2}{7} ln ∣ u - 2 ∣

\int \frac{5}{7 ( u + 5 )} d u = \frac{5}{7} ln ∣ u + 5 ∣

= \frac{2}{7} ln ∣ u - 2 ∣ + \frac{5}{7} ln ∣ u + 5 ∣

Setze in

u = x + 4

ein

= \frac{2}{7} ln ∣ x + 4 - 2 ∣ + \frac{5}{7} ln ∣ x + 4 + 5 ∣

Vereinfache

\frac{2}{7} ln ∣ x + 4 - 2 ∣ + \frac{5}{7} ln ∣ x + 4 + 5 ∣ : \frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x + 9 ∣

= \frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x + 9 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x + 9 ∣ + C

Graph

Beliebte Beispiele

integral of sqrt(4-x^2-y^2)

\int 4 - x^{2} - y^{2} d x

integral of x^2ln(1+x)

\int x^{2} ln (1 + x) d x

integral of 4xcos(4x^2)

\int 4 x cos (4 x^{2}) d x

(dy)/(dx)=(y^3)/(3x-2),y(1)=-1

\frac{d y}{d x} = \frac{y ^{3}}{3 x - 2}, y (1) = - 1

integral from-infinity to 5 of 1/(x^2+1)

\int_{- \infty}^{5} \frac{1}{x ^{2} + 1} d x