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Beliebt Rechnen >

integral of (x^2)/((x-6)^3)

Lösung

\int \frac{x ^{2}}{( x - 6 ) ^{3}} d x

Lösung

ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( x - 6 ) ^{3}} d x

Wende U-Substitution an

= \int \frac{( u + 6 ) ^{2}}{u ^{3}} d u

Multipliziere aus

\frac{( u + 6 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{12}{u ^{2}} + \frac{36}{u ^{3}}

= \int \frac{1}{u} + \frac{12}{u ^{2}} + \frac{36}{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{12}{u ^{2}} d u + \int \frac{36}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

\int \frac{36}{u ^{3}} d u = - \frac{18}{u ^{2}}

= ln ∣ u ∣ - \frac{12}{u} - \frac{18}{u ^{2}}

Setze in

u = x - 6

ein

= ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 6 ∣ - \frac{12}{x - 6} - \frac{18}{( x - 6 ) ^{2}} + C

Graph

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