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Beliebt Rechnen >

integral of ((3x^4-2x^2+2))/(x^3(x-1))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{3 x ^{4}}{x ^{3} ( x - 1 )} d x = 3 (x - 1 + ln ∣ x - 1 ∣)

\int \frac{2 x ^{2}}{x ^{3} ( x - 1 )} d x = 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

\int \frac{2}{x ^{3} ( x - 1 )} d x = - 2 ln ∣ x ∣ + \frac{2}{x} + \frac{1}{x ^{2}} + 2 ln ∣ x - 1 ∣

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (10x^2+5)

\int (10 x^{2} + 5) d x

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\int \frac{1}{x - 2 x ^{3}} d x

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