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Beliebt Rechnen >

integral of ((2x-1))/(((x+1)(x^2+4x+4)))

Lösung

\int \frac{( 2 x - 1 )}{( ( x + 1 ) ( x ^{2} + 4 x + 4 ) )} d x

Lösung

- 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

Schritte zur Lösung

\int \frac{2 x - 1}{( x + 1 ) ( x ^{2} + 4 x + 4 )} d x

Ermittle den Partialbruch von

\frac{2 x - 1}{( x + 1 ) ( x ^{2} + 4 x + 4 )} : - \frac{3}{x + 1} + \frac{3}{x + 2} + \frac{5}{( x + 2 ) ^{2}}

= \int - \frac{3}{x + 1} + \frac{3}{x + 2} + \frac{5}{( x + 2 ) ^{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{3}{x + 1} d x + \int \frac{3}{x + 2} d x + \int \frac{5}{( x + 2 ) ^{2}} d x

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

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

\int \frac{5}{( x + 2 ) ^{2}} d x = - \frac{5}{x + 2}

= - 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2}

Füge eine Konstante zur Lösung hinzu

= - 3 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

Graph

Beliebte Beispiele

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\int - 3 e^{\frac{- x}{3}} d x

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

integral of (x+1)/(x^3+5x^2+6x)

\int \frac{x + 1}{x ^{3} + 5 x ^{2} + 6 x} d x