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Beliebt Rechnen >

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

Lösung

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

Lösung

x + 3 ln ∣ x - 1 ∣ - \frac{3}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3}}{( x - 1 ) ^{3}} : 1 + \frac{3}{x - 1} + \frac{3}{( x - 1 ) ^{2}} + \frac{1}{( x - 1 ) ^{3}}

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

Wende die Summenregel an:

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

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

\int 1 d x = x

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(x^2-x^4)

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

tangent 5x^3-x^2y+y^3+1=0,(1,-2)

t an g e n t 5 x^{3} - x^{2} y + y^{3} + 1 = 0, (1, - 2)

tangent y=t^3-4t-3,(2,-3)

t an g e n t y = t^{3} - 4 t - 3, (2, - 3)

limit as x approaches 2+of 3/((x-2)^3)

x \to 2 + lim (\frac{3}{( x - 2 ) ^{3}})

integral of (ye^y)

\int (y e^{y}) d y