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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{3}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + 3}{x ( x + 2 )} : \frac{x}{x ( x + 2 )} + \frac{3}{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{3}{x ( x + 2 )} d x

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (2*3^x-5*2^x)/(3^x)

\int \frac{2 \cdot 3 ^{x} - 5 \cdot 2 ^{x}}{3 ^{x}} d x

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

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

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

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

integral of (x^2+4x+27)/((x-2)(x^2+9))

\int \frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} d x

integral of 3/(2x^3)

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