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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int u + 3 + \frac{3}{u} + \frac{1}{u ^{2}} 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 u d u + \int 3 d u + \int \frac{3}{u} d u + \int \frac{1}{u ^{2}} d u

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

\int 3 d u = 3 u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

= \frac{u ^{2}}{2} + 3 u + 3 ln ∣ u ∣ - \frac{1}{u}

Setze in

u = x - 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\int 600 - \frac{x}{3} d x

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