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Beliebt Rechnen >

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

Lösung

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

Lösung

- 5 x + 6 ln ∣ x + 4 ∣ + \frac{1}{2} (x + 4)^{2} - 20 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} + 3 x + 2}{x + 4} : \frac{x ^{2}}{x + 4} + \frac{3 x}{x + 4} + \frac{2}{x + 4}

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

\int \frac{x ^{2}}{x + 4} d x = \frac{( x + 4 ) ^{2}}{2} - 8 (x + 4) + 16 ln ∣ x + 4 ∣

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

\int \frac{2}{x + 4} d x = 2 ln ∣ x + 4 ∣

= \frac{( x + 4 ) ^{2}}{2} - 8 (x + 4) + 16 ln ∣ x + 4 ∣ + 3 (x + 4 - 4 ln ∣ x + 4 ∣) + 2 ln ∣ x + 4 ∣

Vereinfache

\frac{( x + 4 ) ^{2}}{2} - 8 (x + 4) + 16 ln ∣ x + 4 ∣ + 3 (x + 4 - 4 ln ∣ x + 4 ∣) + 2 ln ∣ x + 4 ∣ : - 5 x + 6 ln ∣ x + 4 ∣ + \frac{1}{2} (x + 4)^{2} - 20

= - 5 x + 6 ln ∣ x + 4 ∣ + \frac{1}{2} (x + 4)^{2} - 20

Füge eine Konstante zur Lösung hinzu

= - 5 x + 6 ln ∣ x + 4 ∣ + \frac{1}{2} (x + 4)^{2} - 20 + C

Graph

Beliebte Beispiele

integral of (200e^{0.1x}+120e^{-0.3x})

\int (200 e^{0.1 x} + 120 e^{- 0.3 x}) d x

integral of x^3-3x^2+7x^{-2}+e^x

\int x^{3} - 3 x^{2} + 7 x^{- 2} + e^{x} d x

integral of 1/(a*x-x^2)

\int \frac{1}{a \cdot x - x ^{2}} d x

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

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

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

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