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Beliebt Rechnen >

integral of (x+2)/(x^2+5x)

Lösung

\int \frac{x + 2}{x ^{2} + 5 x} d x

Lösung

\frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C

Schritte zur Lösung

\int \frac{x + 2}{x ^{2} + 5 x} d x

Multipliziere aus

\frac{x + 2}{x ^{2} + 5 x} : \frac{x}{x ^{2} + 5 x} + \frac{2}{x ^{2} + 5 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} + 5 x} d x + \int \frac{2}{x ^{2} + 5 x} d x

\int \frac{x}{x ^{2} + 5 x} d x = ln ∣ x + 5 ∣

\int \frac{2}{x ^{2} + 5 x} d x = 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

= ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

Vereinfache

ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣) : \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C

Graph

Beliebte Beispiele

integral from 0 to 6 of 2e^x

\int_{0}^{6} 2 e^{x} d x

derivative of sin(arccos(2x))

\frac{d}{d x} (sin (arccos (2 x)))

derivative of 5cos(4ln(x))

\frac{d}{d x} (5 cos (4 ln (x)))

integral of (7x^6+x/7)

\int (7 x^{6} + \frac{x}{7}) d x

derivative of (arccos(x)^{10})

\frac{d}{d x} ((arccos (x))^{10})