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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sin^2(x)cos^2(3x)

\int sin^{2} (x) cos^{2} (3 x) d x

integral of ln(sin(x))cos(x)

\int ln (sin (x)) cos (x) d x

integral of ln(x+y)

\int ln (x + y) d x

integral of (x^2+1)/(x^3+10x^2+21x)

\int \frac{x ^{2} + 1}{x ^{3} + 10 x ^{2} + 21 x} d x

integral of (11+x)/x

\int \frac{11 + x}{x} d x