integral of (21x-69)/((x-2)(x-5))

Lösung

\int \frac{21 x - 69}{( x - 2 ) ( x - 5 )} d x

9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣ + C

Schritte zur Lösung

\int \frac{21 x - 69}{( x - 2 ) ( x - 5 )} d x

Wende U-Substitution an

= \int \frac{21 u - 27}{u ( u - 3 )} d u

Ermittle den Partialbruch von

\frac{21 u - 27}{u ( u - 3 )} : \frac{9}{u} + \frac{12}{u - 3}

= \int \frac{9}{u} + \frac{12}{u - 3} 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{9}{u} d u + \int \frac{12}{u - 3} d u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

\int \frac{12}{u - 3} d u = 12 ln ∣ u - 3 ∣

= 9 ln ∣ u ∣ + 12 ln ∣ u - 3 ∣

Setze in

u = x - 2

ein

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 2 - 3 ∣

Vereinfache

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 9 ln ∣ x - 2 ∣ + 12 ln ∣ x - 5 ∣ + C

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