integral of (x-1)/((x-2)(x-3))

Lösung

\int \frac{x - 1}{( x - 2 ) ( x - 3 )} d x

2 ln ∣ x - 3 ∣ - ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x - 1}{( x - 2 ) ( x - 3 )} d x

Wende U-Substitution an

= \int \frac{u + 2}{u ( u + 1 )} d u

Ermittle den Partialbruch von

\frac{u + 2}{u ( u + 1 )} : \frac{2}{u} - \frac{1}{u + 1}

= \int \frac{2}{u} - \frac{1}{u + 1} 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}{u} d u - \int \frac{1}{u + 1} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= 2 ln ∣ u ∣ - ln ∣ u + 1 ∣

Setze in

u = x - 3

ein

= 2 ln ∣ x - 3 ∣ - ln ∣ x - 3 + 1 ∣

Vereinfache

= 2 ln ∣ x - 3 ∣ - ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x - 3 ∣ - ln ∣ x - 2 ∣ + C

y^{^{'}} + e^{x} y = 7 e^{x}

d er i v a t i v e (7 x + 1) (x^{2} - 6)

\frac{d}{d y} (a^{2} - y^{2})

\int cos^{3} (2 x) sin^{5} (2 x) d x

\int \frac{5 x}{x ^{2} + 4} d x