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Beliebt Rechnen >

integral of (x+4)/(x^2+9x-10)

Lösung

\int \frac{x + 4}{x ^{2} + 9 x - 10} d x

Lösung

ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣) + C

Schritte zur Lösung

\int \frac{x + 4}{x ^{2} + 9 x - 10} d x

Faktorisiere

x^{2} + 9 x - 10 : (x - 1) (x + 10)

= \int \frac{x + 4}{( x - 1 ) ( x + 10 )} d x

Wende U-Substitution an

= \int \frac{u + 5}{u ( u + 11 )} d u

Multipliziere aus

\frac{u + 5}{u ( u + 11 )} : \frac{u}{u ( u + 11 )} + \frac{5}{u ( u + 11 )}

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{u}{u ( u + 11 )} d u + \int \frac{5}{u ( u + 11 )} d u

\int \frac{u}{u ( u + 11 )} d u = ln ∣ u + 11 ∣

\int \frac{5}{u ( u + 11 )} d u = 5 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

= ln ∣ u + 11 ∣ + 5 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

Setze in

u = x - 1

ein

= ln ∣ x - 1 + 11 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x - 1 + 11 ∣)

Vereinfache

ln ∣ x - 1 + 11 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x - 1 + 11 ∣) : ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣)

= ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣) + C

Graph

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