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Beliebt Rechnen >

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

Lösung

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

Lösung

\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 ^{2} + 2 x + 1} d x

Vervollständige das Quadrat

x^{2} + 2 x + 1 : (x + 1)^{2}

= \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

Graph

Beliebte Beispiele

tangent y=((x^2))/(8+sqrt(x))

t an g e n t y = \frac{( x ^{2} )}{8 + x}

integral of ((1-x^3))/(x\sqrt[3]{x)}

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

integral of ae^{-mx}

\int a e^{- m x} d x

(dy)/(dx)+(3/x)y=-2x+7

\frac{d y}{d x} + (\frac{3}{x}) y = - 2 x + 7

limit as x approaches-1 of 1+2x

x \to - 1 lim (1 + 2 x)