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Beliebt Rechnen >

integral of (3x^2+19x+27)/((x+2)(x+3)^2)

Lösung

\int \frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} d x

Lösung

ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3} + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} : \frac{1}{x + 2} + \frac{2}{x + 3} + \frac{3}{( x + 3 ) ^{2}}

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

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

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

\int \frac{2}{x + 3} d x = 2 ln ∣ x + 3 ∣

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

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3} + C

Graph

Beliebte Beispiele

integral of 10e^{-x/2}

\int 10 e^{- \frac{x}{2}} d x

integral of (1-x)e^{x^2}

\int (1 - x) e^{x^{2}} d x

integral of 1/((sqrt(9-x^2))^3)

\int \frac{1}{( 9 - x ^{2} ) ^{3}} d x

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

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

integral of 2/((x-2)(x-1))

\int \frac{2}{( x - 2 ) ( x - 1 )} d x