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Beliebt Rechnen >

integral of (x-4)/(x^2(x+1))

Lösung

\int \frac{x - 4}{x ^{2} ( x + 1 )} d x

Lösung

5 ln ∣ x ∣ + \frac{4}{x} - 5 ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{x - 4}{x ^{2} ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{x - 4}{x ^{2} ( x + 1 )} : \frac{5}{x} - \frac{4}{x ^{2}} - \frac{5}{x + 1}

= \int \frac{5}{x} - \frac{4}{x ^{2}} - \frac{5}{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{5}{x} d x - \int \frac{4}{x ^{2}} d x - \int \frac{5}{x + 1} d x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int \frac{4}{x ^{2}} d x = - \frac{4}{x}

\int \frac{5}{x + 1} d x = 5 ln ∣ x + 1 ∣

= 5 ln ∣ x ∣ - (- \frac{4}{x}) - 5 ln ∣ x + 1 ∣

Vereinfache

= 5 ln ∣ x ∣ + \frac{4}{x} - 5 ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= 5 ln ∣ x ∣ + \frac{4}{x} - 5 ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of 1/(xlog_{5)(x)}

\int \frac{1}{x lo g _{5} ( x )} d x

integral of x^5(ln(x))^2

\int x^{5} (ln (x))^{2} d x

integral of sec^4(x/5)tan^4(x/5)

\int sec^{4} (\frac{x}{5}) tan^{4} (\frac{x}{5}) d x

integral of (50+y)/(5x+1)

\int \frac{50 + y}{5 x + 1}

integral of (xcot^2(x^2))/(csc(x^2))

\int \frac{x cot ^{2} ( x ^{2} )}{csc ( x ^{2} )} d x