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Beliebt Rechnen >

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

Lösung

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

Lösung

- 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} + 2 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

\int \frac{2}{u - 1} d u = 2 ln ∣ u - 1 ∣

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

Setze in

u = x - 1

ein

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

Vereinfache

= - 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} + 2 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} + 2 ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of (5.4145)/(sqrt(1+0.91t))

\int \frac{5.4145}{1 + 0.91 t} d t

integral of (cos^3(y))/(1+cos(2y))

\int \frac{cos ^{3} ( y )}{1 + cos ( 2 y )} d y

integral of xe^xln(2)

\int x e^{x} ln (2) d x

integral of x-x^{1/2}

\int x - x^{\frac{1}{2}} d x

integral of 21x^2+2

\int 21 x^{2} + 2 d x