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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{2 x ^{3}}{x ^{3} + 1} d x = 2 (- \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x)

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

\int \frac{2}{x ^{3} + 1} d x = 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

= x^{3} + 1 - ln x^{3} + 1 - 2 (- \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x) + \frac{5}{3} ln x^{3} + 1 - 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

Vereinfache

x^{3} + 1 - ln x^{3} + 1 - 2 (- \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x) + \frac{5}{3} ln x^{3} + 1 - 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) : x^{3} - 2 x + \frac{2}{3} ln x^{3} + 1 + 1

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches 4 of (3x)/4+2

x \to 4 lim (\frac{3 x}{4} + 2)

derivative of (x^2-3^3)

\frac{d}{d x} ((x^{2} - 3)^{3})

derivative f(x)=(x-2)(5x+3)

d er i v a t i v e f (x) = (x - 2) (5 x + 3)

tangent f(x)= 1/x ,\at x=-1

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

limit as x approaches 0+of (x)^{sqrt(x)}

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