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Beliebt Rechnen >

integral of (x^2-5x+6)/((2x-1)(x-2)^2)

Lösung

\int \frac{x ^{2} - 5 x + 6}{( 2 x - 1 ) ( x - 2 ) ^{2}} d x

Lösung

\frac{1}{6} (5 ln ∣ 2 x - 1 ∣ - 2 ln ∣ 2 x - 4 ∣) + C

Schritte zur Lösung

\int \frac{x ^{2} - 5 x + 6}{( 2 x - 1 ) ( x - 2 ) ^{2}} d x

Wende U-Substitution an

= \int \frac{u - 5}{2 u ( u - 3 )} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{u - 5}{u ( u - 3 )} d u

Multipliziere aus

\frac{u - 5}{u ( u - 3 )} : \frac{u}{u ( u - 3 )} - \frac{5}{u ( u - 3 )}

Wende die Summenregel an:

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

= \frac{1}{2} (\int \frac{u}{u ( u - 3 )} d u - \int \frac{5}{u ( u - 3 )} d u)

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

\int \frac{5}{u ( u - 3 )} d u = 5 (- \frac{1}{3} ln ∣ u ∣ + \frac{1}{3} ln ∣ u - 3 ∣)

= \frac{1}{2} (ln ∣ u - 3 ∣ - 5 (- \frac{1}{3} ln ∣ u ∣ + \frac{1}{3} ln ∣ u - 3 ∣))

Setze in

u = 2 x - 1

ein

= \frac{1}{2} (ln ∣ 2 x - 1 - 3 ∣ - 5 (- \frac{1}{3} ln ∣ 2 x - 1 ∣ + \frac{1}{3} ln ∣ 2 x - 1 - 3 ∣))

Vereinfache

\frac{1}{2} (ln ∣ 2 x - 1 - 3 ∣ - 5 (- \frac{1}{3} ln ∣ 2 x - 1 ∣ + \frac{1}{3} ln ∣ 2 x - 1 - 3 ∣)) : \frac{1}{6} (5 ln ∣ 2 x - 1 ∣ - 2 ln ∣ 2 x - 4 ∣)

= \frac{1}{6} (5 ln ∣ 2 x - 1 ∣ - 2 ln ∣ 2 x - 4 ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} (5 ln ∣ 2 x - 1 ∣ - 2 ln ∣ 2 x - 4 ∣) + C

Graph

Beliebte Beispiele

limit as x approaches infinity of x^3-3

x \to \infty lim (x^{3} - 3)

derivative of 10cos(2x)

\frac{d}{d x} (10 cos (2 x))

(\partial)/(\partial x)(2(x+1))

\frac{\partial}{\partial x} (2 (x + 1))

integral of 9/(x(x^2+1))

\int \frac{9}{x ( x ^{2} + 1 )} d x

integral of x^2-2x+3

\int x^{2} - 2 x + 3 d x