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Beliebt Rechnen >

integral of (x^2+3x+2)/(x^4-x^2)

Lösung

\int \frac{x ^{2} + 3 x + 2}{x ^{4} - x ^{2}} d x

Lösung

- 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x} + C

Schritte zur Lösung

\int \frac{x ^{2} + 3 x + 2}{x ^{4} - x ^{2}} d x

Vereinfache

\frac{x ^{2} + 3 x + 2}{x ^{4} - x ^{2}} : \frac{x + 2}{x ^{2} ( x - 1 )}

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

Multipliziere aus

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

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

\int \frac{x}{x ^{2} ( x - 1 )} d x = - ln ∣ x ∣ + ln ∣ x - 1 ∣

\int \frac{2}{x ^{2} ( x - 1 )} d x = 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

= - ln ∣ x ∣ + ln ∣ x - 1 ∣ + 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

Vereinfache

- ln ∣ x ∣ + ln ∣ x - 1 ∣ + 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x}

= - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x}

Füge eine Konstante zur Lösung hinzu

= - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x} + C

Graph

Beliebte Beispiele

integral of 15r^2

\int 15 r^{2} d r

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\int - sin (x) sin (y) d x

integral of 5sin(2x)e^x

\int 5 sin (2 x) e^{x} d x

integral of ((ln(x))^2)/(4x^3)

\int \frac{( ln ( x ) ) ^{2}}{4 x ^{3}} d x

integral of (3x^3)/(x^4-1)

\int \frac{3 x ^{3}}{x ^{4} - 1} d x