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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ + ln ∣ 2 + x ∣ + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Multipliziere aus

\frac{1 + x}{x ( 2 + x )} : \frac{1}{x ( 2 + x )} + \frac{x}{x ( 2 + x )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Vereinfache

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

= ln ∣ x ∣ + ln ∣ 2 + x ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + ln ∣ 2 + x ∣ + C

Graph

Beliebte Beispiele

integral of x^5*e^x

\int x^{5} \cdot e^{x} d x

integral of (t+1)/t

\int \frac{t + 1}{t} d t

limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2)

(x, y) \to (0, 0) lim ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

steigung y=4x-x^2

s l o p e y = 4 x - x^{2}

derivative of x/3 (2x+1^3)

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