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Beliebt Rechnen >

integral of (6(1+x))/(x(x^2+7x+6))

Lösung

\int \frac{6 ( 1 + x )}{x ( x ^{2} + 7 x + 6 )} d x

Lösung

ln ∣ x ∣ - ln ∣ x + 6 ∣ + C

Schritte zur Lösung

\int \frac{6 ( 1 + x )}{x ( x ^{2} + 7 x + 6 )} 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}{x ( x ^{2} + 7 x + 6 )} d x

Ermittle den Partialbruch von

\frac{1 + x}{x ( x ^{2} + 7 x + 6 )} : \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

(\partial)/(\partial x)(cos^4(x^3y^5))

\frac{\partial}{\partial x} (cos^{4} (x^{3} y^{5}))

limit as x approaches-8 of x^2-3

x \to - 8 lim (x^{2} - 3)

derivative f(x)=4x^3-8x

d er i v a t i v e f (x) = 4 x^{3} - 8 x

derivative ae^{bt}

d er i v a t i v e a e^{b t}

tangent f(x)= 3/(x+2),\at x=3

t an g e n t f (x) = \frac{3}{x + 2}, at x = 3