de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 4/(x^3-8)

Lösung

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

Lösung

\frac{1}{3} ln ∣ x - 2 ∣ - \frac{1}{6} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= 4 \cdot \int \frac{1}{x ^{3} - 8} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} - 8} : \frac{1}{12 ( x - 2 )} + \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )}

= 4 \cdot \int \frac{1}{12 ( x - 2 )} + \frac{- x - 4}{12 ( x ^{2} + 2 x + 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

= 4 (\int \frac{1}{12 ( x - 2 )} d x + \int \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )} d x)

\int \frac{1}{12 ( x - 2 )} d x = \frac{1}{12} ln ∣ x - 2 ∣

\int \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )} d x = \frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1)))

= 4 (\frac{1}{12} ln ∣ x - 2 ∣ + \frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1))))

Vereinfache

4 (\frac{1}{12} ln ∣ x - 2 ∣ + \frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1)))) : \frac{1}{3} ln ∣ x - 2 ∣ - \frac{1}{6} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

= \frac{1}{3} ln ∣ x - 2 ∣ - \frac{1}{6} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x - 2 ∣ - \frac{1}{6} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Graph

Beliebte Beispiele

integral of 1/(sqrt(9y^2+6y+5))

\int \frac{1}{9 y ^{2} + 6 y + 5} d y

integral of cos(pi/2 x^2)

\int cos (\frac{π}{2} x^{2}) d x

integral of 1/(csc(x)+cot(x))

\int \frac{1}{csc ( x ) + cot ( x )} d x

integral of (4x+1)cos(x/3)

\int (4 x + 1) cos (\frac{x}{3}) d x

integral of (3x^2+9x^4-x)/(x^2)

\int \frac{3 x ^{2} + 9 x ^{4} - x}{x ^{2}} d x