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Beliebt Rechnen >

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

Lösung

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

Lösung

- x + \frac{1}{3} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 1 + 3 arctan (x)) + \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{3 x ^{5} + 1}{3 x - 3 x ^{5}} : - 1 + \frac{1}{3 x} + \frac{- x + 3}{6 ( x ^{2} + 1 )} + \frac{1}{6 ( x + 1 )} - \frac{1}{3 ( x - 1 )}

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

\int 1 d x = x

\int \frac{1}{3 x} d x = \frac{1}{3} ln ∣ x ∣

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

\int \frac{1}{6 ( x + 1 )} d x = \frac{1}{6} ln ∣ x + 1 ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative ln(1-x^2)

d er i v a t i v e ln (1 - x^{2})

derivative x^{-1/2}

d er i v a t i v e x^{- \frac{1}{2}}

fläche y=3-x,y=2x^2,[0,1]

a re a y = 3 - x, y = 2 x^{2}, [0, 1]

tangent f(x)=2x^2+3x,(-2,2)

t an g e n t f (x) = 2 x^{2} + 3 x, (- 2, 2)

(\partial)/(\partial x)(x^3-12x+y^3-27y)

\frac{\partial}{\partial x} (x^{3} - 12 x + y^{3} - 27 y)