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Beliebt Rechnen >

integral of (9x+15)/(x^2+2x+1)

Lösung

\int \frac{9 x + 15}{x ^{2} + 2 x + 1} d x

Lösung

9 ln ∣ x + 1 ∣ - \frac{6}{x + 1} + C

Schritte zur Lösung

\int \frac{9 x + 15}{x ^{2} + 2 x + 1} d x

Vervollständige das Quadrat

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

= \int \frac{9 x + 15}{( x + 1 ) ^{2}} d x

Wende U-Substitution an

= \int \frac{9 u + 6}{u ^{2}} d u

Multipliziere aus

\frac{9 u + 6}{u ^{2}} : \frac{9}{u} + \frac{6}{u ^{2}}

= \int \frac{9}{u} + \frac{6}{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{9}{u} d u + \int \frac{6}{u ^{2}} d u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

\int \frac{6}{u ^{2}} d u = - \frac{6}{u}

= 9 ln ∣ u ∣ - \frac{6}{u}

Setze in

u = x + 1

ein

= 9 ln ∣ x + 1 ∣ - \frac{6}{x + 1}

Füge eine Konstante zur Lösung hinzu

= 9 ln ∣ x + 1 ∣ - \frac{6}{x + 1} + C

Graph

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