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Beliebt Rechnen >

integral of 6/(x^2+6x)

Lösung

\int \frac{6}{x ^{2} + 6 x} d x

Lösung

ln ∣ x ∣ - ln ∣ x + 6 ∣ + C

Schritte zur Lösung

\int \frac{6}{x ^{2} + 6 x} d x

Entferne die Konstante:

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

= 6 \cdot \int \frac{1}{x ^{2} + 6 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 6 x} : \frac{1}{6 x} - \frac{1}{6 ( x + 6 )}

= 6 \cdot \int \frac{1}{6 x} - \frac{1}{6 ( x + 6 )} d x

Wende die Summenregel an:

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

= 6 (\int \frac{1}{6 x} d x - \int \frac{1}{6 ( x + 6 )} d x)

\int \frac{1}{6 x} d x = \frac{1}{6} ln ∣ x ∣

\int \frac{1}{6 ( x + 6 )} d x = \frac{1}{6} ln ∣ x + 6 ∣

= 6 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x + 6 ∣)

Vereinfache

6 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x + 6 ∣) : ln ∣ x ∣ - ln ∣ x + 6 ∣

= ln ∣ x ∣ - ln ∣ x + 6 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - ln ∣ x + 6 ∣ + C

Graph

Beliebte Beispiele

integral of 1/(x(ln(x^3)))

\int \frac{1}{x ( ln ( x ^{3} ) )} d x

sum from n=1 to infinity of 5/(n^2+64)

n = 1 \sum \infty \frac{5}{n ^{2} + 64}

integral from 1 to 6x of 7/t

\int_{1}^{6 x} \frac{7}{t} d t

integral of cot^5(x)

\int cot^{5} (x) d x

derivative of (x+4/x)

\frac{d}{d x} (\frac{x + 4}{x})