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Beliebt Rechnen >

integral of 1/(x^7-4x^5+6x^3-4x)

Lösung

\int \frac{1}{x ^{7} - 4 x ^{5} + 6 x ^{3} - 4 x} d x

Lösung

- \frac{1}{4} ln ∣ x ∣ + \frac{1}{8} ln x + 2 + \frac{1}{8} ln x - 2 + \frac{1}{4 x ^{2}} + C

Schritte zur Lösung

\int \frac{1}{x ^{7} - 4 x ^{5} + 6 x ^{3} - 4 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{7} - 4 x ^{5} + 6 x ^{3} - 4 x} : - \frac{1}{4 x} + \frac{1}{8 ( x + 2 )} + \frac{1}{8 ( x - 2 )} - \frac{x}{2 ( x ^{4} - 2 x ^{2} + 2 )}

= \int - \frac{1}{4 x} + \frac{1}{8 ( x + 2 )} + \frac{1}{8 ( x - 2 )} - \frac{x}{2 ( x ^{4} - 2 x ^{2} + 2 )} 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}{4 x} d x + \int \frac{1}{8 ( x + 2 )} d x + \int \frac{1}{8 ( x - 2 )} d x - \int \frac{x}{2 ( x ^{4} - 2 x ^{2} + 2 )} d x

\int \frac{1}{4 x} d x = \frac{1}{4} ln ∣ x ∣

\int \frac{1}{8 ( x + 2 )} d x = \frac{1}{8} ln x + 2

\int \frac{1}{8 ( x - 2 )} d x = \frac{1}{8} ln x - 2

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

= - \frac{1}{4} ln ∣ x ∣ + \frac{1}{8} ln x + 2 + \frac{1}{8} ln x - 2 - (- \frac{1}{4 x ^{2}})

Vereinfache

= - \frac{1}{4} ln ∣ x ∣ + \frac{1}{8} ln x + 2 + \frac{1}{8} ln x - 2 + \frac{1}{4 x ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} ln ∣ x ∣ + \frac{1}{8} ln x + 2 + \frac{1}{8} ln x - 2 + \frac{1}{4 x ^{2}} + C

Graph

Beliebte Beispiele

integral of 3^x2^x

\int 3^{x} 2^{x} d x

integral of 5/(4-sqrt(3-z))

\int \frac{5}{4 - 3 - z} d z

integral of 2/(xsqrt(x-2))

\int \frac{2}{x x - 2} d x

integral of (16t^2+16t+4)

\int (16 t^{2} + 16 t + 4) d t

integral of (6x-5)/(sqrt(2-3x))

\int \frac{6 x - 5}{2 - 3 x} d x