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Beliebt Rechnen >

integral of (x-4)/(x^2+3x+2)

Lösung

\int \frac{x - 4}{x ^{2} + 3 x + 2} d x

Lösung

- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ + 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣) + C

Schritte zur Lösung

\int \frac{x - 4}{x ^{2} + 3 x + 2} d x

Multipliziere aus

\frac{x - 4}{x ^{2} + 3 x + 2} : \frac{x}{x ^{2} + 3 x + 2} - \frac{4}{x ^{2} + 3 x + 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} + 3 x + 2} d x - \int \frac{4}{x ^{2} + 3 x + 2} d x

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

\int \frac{4}{x ^{2} + 3 x + 2} d x = - 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣)

= - ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ - (- 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣))

Vereinfache

= - ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ + 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣)

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ + 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣) + C

Graph

Beliebte Beispiele

integral of (x-x^3)/x

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

integral of 2-e^{-x}

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

integral of 16tan^2(x)sec(x)

\int 16 tan^{2} (x) sec (x) d x

integral of (2sec^2(x)-8sec^2(5x))

\int (2 sec^{2} (x) - 8 sec^{2} (5 x)) d x

integral of 1/(sqrt(9-x^2-8x))

\int \frac{1}{9 - x ^{2} - 8 x} d x