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Beliebt Rechnen >

integral of ((5x-7))/((x^2-x-2)(x-3))

Lösung

\int \frac{( 5 x - 7 )}{( x ^{2} - x - 2 ) ( x - 3 )} d x

Lösung

- ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣ + 2 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{5 x - 7}{( x ^{2} - x - 2 ) ( x - 3 )} d x

Ermittle den Partialbruch von

\frac{5 x - 7}{( x ^{2} - x - 2 ) ( x - 3 )} : - \frac{1}{x + 1} - \frac{1}{x - 2} + \frac{2}{x - 3}

= \int - \frac{1}{x + 1} - \frac{1}{x - 2} + \frac{2}{x - 3} 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}{x + 1} d x - \int \frac{1}{x - 2} d x + \int \frac{2}{x - 3} d x

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

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

\int \frac{2}{x - 3} d x = 2 ln ∣ x - 3 ∣

= - ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣ + 2 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣ + 2 ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

derivative y=(5x-4)^5(3-x^3)^5

tangent f(x)=2x^2-6x,\at x=1

tangent g(x)=sqrt(5x+20),\at x=1

derivative sqrt(4x+3)