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Beliebt Rechnen >

integral of 1/(x^3-16x)

Lösung

\int \frac{1}{x ^{3} - 16 x} d x

Lösung

- \frac{1}{16} ln ∣ x ∣ + \frac{1}{32} ln ∣ x + 4 ∣ + \frac{1}{32} ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{3} - 16 x} d x

Faktorisiere

x^{3} - 16 x : - (- x^{3} + 16 x)

= \int \frac{1}{- ( - x ^{3} + 16 x )} d x

Entferne die Konstante:

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

= - \int \frac{1}{- x ^{3} + 16 x} d x

Ermittle den Partialbruch von

\frac{1}{- x ^{3} + 16 x} : \frac{1}{16 x} - \frac{1}{32 ( x + 4 )} - \frac{1}{32 ( x - 4 )}

= - \int \frac{1}{16 x} - \frac{1}{32 ( x + 4 )} - \frac{1}{32 ( 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

= - (\int \frac{1}{16 x} d x - \int \frac{1}{32 ( x + 4 )} d x - \int \frac{1}{32 ( x - 4 )} d x)

\int \frac{1}{16 x} d x = \frac{1}{16} ln ∣ x ∣

\int \frac{1}{32 ( x + 4 )} d x = \frac{1}{32} ln ∣ x + 4 ∣

\int \frac{1}{32 ( x - 4 )} d x = \frac{1}{32} ln ∣ x - 4 ∣

= - (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣)

Vereinfache

= - \frac{1}{16} ln ∣ x ∣ + \frac{1}{32} ln ∣ x + 4 ∣ + \frac{1}{32} ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{16} ln ∣ x ∣ + \frac{1}{32} ln ∣ x + 4 ∣ + \frac{1}{32} ln ∣ x - 4 ∣ + C

Graph

Beliebte Beispiele

integral of 7ln(2x)

\int 7 ln (2 x) d x

integral of x^2(1/(2x))

\int x^{2} (\frac{1}{2 x}) d x

integral of (6y-7)/((2y+1)(4y^2+1))

\int \frac{6 y - 7}{( 2 y + 1 ) ( 4 y ^{2} + 1 )} d y

integral of (7x^2)/((144+x^2)^2)

\int \frac{7 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

integral of (y+4)/(y-4)

\int \frac{y + 4}{y - 4} d y