integral of (-8x+5)/(x^3+4x)

Lösung

\int \frac{- 8 x + 5}{x ^{3} + 4 x} d x

\frac{5}{4} ln ∣ x ∣ - \frac{1}{4} (\frac{5}{2} ln \frac{x ^{2}}{4} + 1 + 16 arctan (\frac{x}{2})) + C

Schritte zur Lösung

\int \frac{- 8 x + 5}{x ^{3} + 4 x} d x

Ermittle den Partialbruch von

\frac{- 8 x + 5}{x ^{3} + 4 x} : \frac{5}{4 x} + \frac{- 5 x - 32}{4 ( x ^{2} + 4 )}

= \int \frac{5}{4 x} + \frac{- 5 x - 32}{4 ( x ^{2} + 4 )} 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{5}{4 x} d x + \int \frac{- 5 x - 32}{4 ( x ^{2} + 4 )} d x

\int \frac{5}{4 x} d x = \frac{5}{4} ln ∣ x ∣

\int \frac{- 5 x - 32}{4 ( x ^{2} + 4 )} d x = - \frac{1}{4} (\frac{5}{2} ln \frac{x ^{2}}{4} + 1 + 16 arctan (\frac{x}{2}))

= \frac{5}{4} ln ∣ x ∣ - \frac{1}{4} (\frac{5}{2} ln \frac{x ^{2}}{4} + 1 + 16 arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{5}{4} ln ∣ x ∣ - \frac{1}{4} (\frac{5}{2} ln \frac{x ^{2}}{4} + 1 + 16 arctan (\frac{x}{2})) + C

x \to 36 lim (x^{\frac{3}{2}})

y^{^{''}} + x = 7 y

\frac{\partial}{\partial y} (2 x e^{- 2 x y})

\frac{\partial}{\partial x} (z^{2} x)

p a r i t y x + x + x