integral of ((x^2-x+18))/(x^3+3x)

Lösung

\int \frac{( x ^{2} - x + 18 )}{x ^{3} + 3 x} d x

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

Schritte zur Lösung

\int \frac{x ^{2} - x + 18}{x ^{3} + 3 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 18}{x ^{3} + 3 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 3}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 3} 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 3} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 3} d x = - \frac{5}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

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

Füge eine Konstante zur Lösung hinzu

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

\int (3 x^{4}) d x

t an g e n t f (x) = x - 3, at x = 7

2 t y^{^{'}} = 8 y

\frac{d}{d x} (8 ln (sin (x)))

\int e^{6 x} (6) d x