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Beliebt Rechnen >

integral of (x^3)/((x+3)^2)

Lösung

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

Lösung

\frac{1}{2} (x + 3)^{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ + \frac{27}{x + 3} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 3 ) ^{3}}{u ^{2}} : u - 9 + \frac{27}{u} - \frac{27}{u ^{2}}

= \int u - 9 + \frac{27}{u} - \frac{27}{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 9 d u + \int \frac{27}{u} d u - \int \frac{27}{u ^{2}} d u

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

\int 9 d u = 9 u

\int \frac{27}{u} d u = 27 ln ∣ u ∣

\int \frac{27}{u ^{2}} d u = - \frac{27}{u}

= \frac{u ^{2}}{2} - 9 u + 27 ln ∣ u ∣ - (- \frac{27}{u})

Setze in

u = x + 3

ein

= \frac{( x + 3 ) ^{2}}{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ - (- \frac{27}{x + 3})

Vereinfache

\frac{( x + 3 ) ^{2}}{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ - (- \frac{27}{x + 3}) : \frac{1}{2} (x + 3)^{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ + \frac{27}{x + 3}

= \frac{1}{2} (x + 3)^{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ + \frac{27}{x + 3}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x + 3)^{2} - 9 (x + 3) + 27 ln ∣ x + 3 ∣ + \frac{27}{x + 3} + C

Graph

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