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Beliebt Rechnen >

integral of (x(x-8))/((x-4)^3)

Lösung

\int \frac{x ( x - 8 )}{( x - 4 ) ^{3}} d x

Lösung

ln ∣ x - 4 ∣ + \frac{8}{( x - 4 ) ^{2}} + C

Schritte zur Lösung

\int \frac{x ( x - 8 )}{( x - 4 ) ^{3}} d x

Vervollständige das Quadrat

x (x - 8) : (x - 4)^{2} - 16

= \int \frac{( x - 4 ) ^{2} - 16}{( x - 4 ) ^{3}} d x

Wende U-Substitution an

= \int \frac{u ^{2} - 16}{u ^{3}} d u

Multipliziere aus

\frac{u ^{2} - 16}{u ^{3}} : \frac{1}{u} - \frac{16}{u ^{3}}

= \int \frac{1}{u} - \frac{16}{u ^{3}} 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{1}{u} d u - \int \frac{16}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{16}{u ^{3}} d u = - \frac{8}{u ^{2}}

= ln ∣ u ∣ - (- \frac{8}{u ^{2}})

Setze in

u = x - 4

ein

= ln ∣ x - 4 ∣ - (- \frac{8}{( x - 4 ) ^{2}})

Vereinfache

= ln ∣ x - 4 ∣ + \frac{8}{( x - 4 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 4 ∣ + \frac{8}{( x - 4 ) ^{2}} + C

Graph

Beliebte Beispiele

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\int \frac{- x ^{2} + 2 x - 8}{( x ^{2} + 4 ) ( x - 2 )} d x

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\int (x^{2} + 10 x) cos (x) d x

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