integral of (11)/(x^3-125)

Lösung

\int \frac{11}{x ^{3} - 125} d x

11 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

Schritte zur Lösung

\int \frac{11}{x ^{3} - 125} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 11 \cdot \int \frac{1}{x ^{3} - 125} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} - 125} : \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )}

= 11 \cdot \int \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} 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

= 11 (\int \frac{1}{75 ( x - 5 )} d x + \int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x)

\int \frac{1}{75 ( x - 5 )} d x = \frac{1}{75} ln ∣ x - 5 ∣

\int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x = \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3}))

= 11 (\frac{1}{75} ln ∣ x - 5 ∣ + \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})))

Multipliziere aus

\frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})) : - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})

= 11 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3}))

Füge eine Konstante zur Lösung hinzu

= 11 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

d er i v a t i v e y = ln (6 - 7 x)

\frac{d}{d x} (cos^{2} (7 x))

(y + 4) y^{^{'}} = y (2 x + 4)

l o n g d i v i s i o n \frac{3 x - 1}{2 x + 1}

t an g e n t x (1.1)