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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (1-sin(sqrt(x)))/(sqrt(x))

\int \frac{1 - sin ( x )}{x} d x

integral of (5x-5)/(x^2-x-6)

\int \frac{5 x - 5}{x ^{2} - x - 6} d x

integral of 2^{1/x}

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

integral of x^7ln(9x^4)

\int x^{7} ln (9 x^{4}) d x

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

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