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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{4}{x + 1} - 5 ln ∣ x + 2 ∣ + 5 ln ∣ x + 1 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

= \frac{4}{x + 1} - 5 ln ∣ x + 2 ∣ + 5 ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{x + 1} - 5 ln ∣ x + 2 ∣ + 5 ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of (x+2y)/((x+y)^2)

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

integral of (x^4-5x)^2(4x^3-5)

\int (x^{4} - 5 x)^{2} (4 x^{3} - 5) d x

integral of (2x)/(sqrt(2-x^2))

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

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

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

integral of 4(2x-3)^3

\int 4 (2 x - 3)^{3} d x