integral of (14x-46)/((x-2)(x-5))

Lösung

\int \frac{14 x - 46}{( x - 2 ) ( x - 5 )} d x

6 ln ∣ x - 2 ∣ + 8 ln ∣ x - 5 ∣ + C

Schritte zur Lösung

\int \frac{14 x - 46}{( x - 2 ) ( x - 5 )} d x

Wende U-Substitution an

= \int \frac{14 u - 18}{u ( u - 3 )} d u

Ermittle den Partialbruch von

\frac{14 u - 18}{u ( u - 3 )} : \frac{6}{u} + \frac{8}{u - 3}

= \int \frac{6}{u} + \frac{8}{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{6}{u} d u + \int \frac{8}{u - 3} d u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

\int \frac{8}{u - 3} d u = 8 ln ∣ u - 3 ∣

= 6 ln ∣ u ∣ + 8 ln ∣ u - 3 ∣

Setze in

u = x - 2

ein

= 6 ln ∣ x - 2 ∣ + 8 ln ∣ x - 2 - 3 ∣

Vereinfache

= 6 ln ∣ x - 2 ∣ + 8 ln ∣ x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x - 2 ∣ + 8 ln ∣ x - 5 ∣ + C

\int (- 3 x + 5)^{3} d x

\int \frac{e ^{5 x} - 2 e ^{4 x} - 1}{e ^{4 x}} d x

\int e^{- (x - 8)^{3}} (x - 8)^{2} d x

\int \frac{x ^{4} + x}{x ^{3} - x ^{2} + x - 1} d x

\int 9 x sin (2 x) d x