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

Lösung

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

\frac{1}{2} (x + 1)^{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})

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (A (e^{- 2 x} - 2 e^{- 2 x} x))

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