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Beliebt Rechnen >

integral of 1/((x-6)(x-3)(x+1))

Lösung

\int \frac{1}{( x - 6 ) ( x - 3 ) ( x + 1 )} d x

Lösung

\frac{1}{21} ln ∣ x - 6 ∣ - \frac{1}{12} ln ∣ x - 3 ∣ + \frac{1}{28} ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{1}{( x - 6 ) ( x - 3 ) ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 6 ) ( x - 3 ) ( x + 1 )} : \frac{1}{21 ( x - 6 )} - \frac{1}{12 ( x - 3 )} + \frac{1}{28 ( x + 1 )}

= \int \frac{1}{21 ( x - 6 )} - \frac{1}{12 ( x - 3 )} + \frac{1}{28 ( x + 1 )} 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

= \int \frac{1}{21 ( x - 6 )} d x - \int \frac{1}{12 ( x - 3 )} d x + \int \frac{1}{28 ( x + 1 )} d x

\int \frac{1}{21 ( x - 6 )} d x = \frac{1}{21} ln ∣ x - 6 ∣

\int \frac{1}{12 ( x - 3 )} d x = \frac{1}{12} ln ∣ x - 3 ∣

\int \frac{1}{28 ( x + 1 )} d x = \frac{1}{28} ln ∣ x + 1 ∣

= \frac{1}{21} ln ∣ x - 6 ∣ - \frac{1}{12} ln ∣ x - 3 ∣ + \frac{1}{28} ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{21} ln ∣ x - 6 ∣ - \frac{1}{12} ln ∣ x - 3 ∣ + \frac{1}{28} ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

(\partial)/(\partial x)(-9x^2y^2-cos(y))

\frac{\partial}{\partial x} (- 9 x^{2} y^{2} - cos (y))

t an g e n t x^{3} + 8

fläche 1/(18x^2),x, x/8

a re a \frac{1}{18 x ^{2}}, x, \frac{x}{8}

(d^2)/(dx^2)(e^{-6x})

\frac{d ^{2}}{d x ^{2}} (e^{- 6 x})

tangent f(x)=3e^xcos(x),\at x=0

t an g e n t f (x) = 3 e^{x} cos (x), at x = 0