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Beliebt Rechnen >

integral of (2x^2-6)/(x^3+x)

Lösung

\int \frac{2 x ^{2} - 6}{x ^{3} + x} d x

Lösung

4 ln x^{2} + 1 - 6 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 6}{x ^{3} + x} d x

Multipliziere aus

\frac{2 x ^{2} - 6}{x ^{3} + x} : \frac{2 x ^{2}}{x ^{3} + x} - \frac{6}{x ^{3} + 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{2 x ^{2}}{x ^{3} + x} d x - \int \frac{6}{x ^{3} + x} d x

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

\int \frac{6}{x ^{3} + x} d x = 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

= ln x^{2} + 1 - 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Vereinfache

ln x^{2} + 1 - 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) : 4 ln x^{2} + 1 - 6 ln ∣ x ∣

= 4 ln x^{2} + 1 - 6 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 4 ln x^{2} + 1 - 6 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of (x^2sqrt(x^3+5))

\int (x^{2} x^{3} + 5) d x

steigung (3,4),(8,-3)

s l o p e (3, 4), (8, - 3)

limit as x approaches 3 of 2x^2+x-7

x \to 3 lim (2 x^{2} + x - 7)

integral of sqrt(x^2-36)

\int x^{2} - 36 d x

geradzahligkeit ln(sqrt((1-sin(x))/(1+sin(x))))

p a r i t y ln (\frac{1 - sin ( x )}{1 + sin ( x )})