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Beliebt Rechnen >

integral of (12x+20)/(x^2+2x+1)

Lösung

\int \frac{12 x + 20}{x ^{2} + 2 x + 1} d x

Lösung

12 ln ∣ x + 1 ∣ - \frac{8}{x + 1} + C

Schritte zur Lösung

\int \frac{12 x + 20}{x ^{2} + 2 x + 1} d x

Vervollständige das Quadrat

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

= \int \frac{12 x + 20}{( x + 1 ) ^{2}} d x

Wende U-Substitution an

= \int \frac{12 u + 8}{u ^{2}} d u

Multipliziere aus

\frac{12 u + 8}{u ^{2}} : \frac{12}{u} + \frac{8}{u ^{2}}

= \int \frac{12}{u} + \frac{8}{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 \frac{12}{u} d u + \int \frac{8}{u ^{2}} d u

\int \frac{12}{u} d u = 12 ln ∣ u ∣

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

= 12 ln ∣ u ∣ - \frac{8}{u}

Setze in

u = x + 1

ein

= 12 ln ∣ x + 1 ∣ - \frac{8}{x + 1}

Füge eine Konstante zur Lösung hinzu

= 12 ln ∣ x + 1 ∣ - \frac{8}{x + 1} + C

Graph

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