integral of (20)/((x-1)*(x^2+9))

Lösung

\int \frac{20}{( x - 1 ) \cdot ( x ^{2} + 9 )} d x

20 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}))) + C

Schritte zur Lösung

\int \frac{20}{( x - 1 ) ( x ^{2} + 9 )} d x

Entferne die Konstante:

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

= 20 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 9 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 1 ) ( x ^{2} + 9 )} : \frac{1}{10 ( x - 1 )} + \frac{- x - 1}{10 ( x ^{2} + 9 )}

= 20 \cdot \int \frac{1}{10 ( x - 1 )} + \frac{- x - 1}{10 ( x ^{2} + 9 )} 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

= 20 (\int \frac{1}{10 ( x - 1 )} d x + \int \frac{- x - 1}{10 ( x ^{2} + 9 )} d x)

\int \frac{1}{10 ( x - 1 )} d x = \frac{1}{10} ln ∣ x - 1 ∣

\int \frac{- x - 1}{10 ( x ^{2} + 9 )} d x = \frac{1}{10} (- \frac{1}{2} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}))

= 20 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})))

Füge eine Konstante zur Lösung hinzu

= 20 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}))) + C

y^{3} - 2 x - 2 d x + 3 y^{2} x d y = 0

\frac{d}{d x} (sec (x) tan (x) (x) (x + 1))

\frac{\partial}{\partial x} (8 x e^{x y})

n = 1 \sum \infty \frac{n !}{n 2 ^{n}}

\frac{d y}{d x} = \frac{1}{y - x}