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Beliebt Rechnen >

integral of 1/(x^4-81)

Lösung

\int \frac{1}{x ^{4} - 81} d x

Lösung

- \frac{1}{54} arctan (\frac{x}{3}) - \frac{1}{108} ln ∣ x + 3 ∣ + \frac{1}{108} ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{4} - 81} d x

Ermittle den Partialbruch von

\frac{1}{x ^{4} - 81} : - \frac{1}{18 ( x ^{2} + 9 )} - \frac{1}{108 ( x + 3 )} + \frac{1}{108 ( x - 3 )}

= \int - \frac{1}{18 ( x ^{2} + 9 )} - \frac{1}{108 ( x + 3 )} + \frac{1}{108 ( x - 3 )} 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}{18 ( x ^{2} + 9 )} d x - \int \frac{1}{108 ( x + 3 )} d x + \int \frac{1}{108 ( x - 3 )} d x

\int \frac{1}{18 ( x ^{2} + 9 )} d x = \frac{1}{54} arctan (\frac{x}{3})

\int \frac{1}{108 ( x + 3 )} d x = \frac{1}{108} ln ∣ x + 3 ∣

\int \frac{1}{108 ( x - 3 )} d x = \frac{1}{108} ln ∣ x - 3 ∣

= - \frac{1}{54} arctan (\frac{x}{3}) - \frac{1}{108} ln ∣ x + 3 ∣ + \frac{1}{108} ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{54} arctan (\frac{x}{3}) - \frac{1}{108} ln ∣ x + 3 ∣ + \frac{1}{108} ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

integral of 1/(3x^2-2x+60)

\int \frac{1}{3 x ^{2} - 2 x + 60} d x

integral of x/(e^{7x^2)}

\int \frac{x}{e ^{7 x^{2}}} d x

integral of 12y^2(6y^3-7)^4

\int 12 y^{2} (6 y^{3} - 7)^{4} d y

integral of 3/(xln(8x))

\int \frac{3}{x ln ( 8 x )} d x

integral of (x^2+2x+1)/(x^3+3x^2+3x)

\int \frac{x ^{2} + 2 x + 1}{x ^{3} + 3 x ^{2} + 3 x} d x