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Beliebt Rechnen >

integral of (5x^2+2x+5)/(x^3-1)

Lösung

\int \frac{5 x ^{2} + 2 x + 5}{x ^{3} - 1} d x

Lösung

4 ln ∣ x - 1 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Schritte zur Lösung

\int \frac{5 x ^{2} + 2 x + 5}{x ^{3} - 1} d x

Ermittle den Partialbruch von

\frac{5 x ^{2} + 2 x + 5}{x ^{3} - 1} : \frac{4}{x - 1} + \frac{x - 1}{x ^{2} + x + 1}

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

\int \frac{4}{x - 1} d x = 4 ln ∣ x - 1 ∣

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

= 4 ln ∣ x - 1 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

Füge eine Konstante zur Lösung hinzu

= 4 ln ∣ x - 1 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Graph

Beliebte Beispiele

limit as x approaches 0 of 8^x

x \to 0 lim (8^{x})

derivative f(x)=sqrt(2)

d er i v a t i v e f (x) = 2

integral of (ln(12x)-ln(4x))

\int (ln (12 x) - ln (4 x)) d x

tangent f(x)=(-8)/(x-1)

t an g e n t f (x) = \frac{- 8}{x - 1}

y^{''}+2y^'-3y=0,y(0)=6,y^'(0)=-2

y^{^{''}} + 2 y^{^{'}} - 3 y = 0, y (0) = 6, y^{^{'}} (0) = - 2