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Beliebt Rechnen >

integral of 1/(16-x^4)

Lösung

\int \frac{1}{16 - x ^{4}} d x

Lösung

\frac{1}{16} arctan (\frac{x}{2}) + \frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{32} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{1}{16 - x ^{4}} d x

Ermittle den Partialbruch von

\frac{1}{16 - x ^{4}} : \frac{1}{8 ( x ^{2} + 4 )} + \frac{1}{32 ( x + 2 )} - \frac{1}{32 ( x - 2 )}

= \int \frac{1}{8 ( x ^{2} + 4 )} + \frac{1}{32 ( x + 2 )} - \frac{1}{32 ( x - 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}{8 ( x ^{2} + 4 )} d x + \int \frac{1}{32 ( x + 2 )} d x - \int \frac{1}{32 ( x - 2 )} d x

\int \frac{1}{8 ( x ^{2} + 4 )} d x = \frac{1}{16} arctan (\frac{x}{2})

\int \frac{1}{32 ( x + 2 )} d x = \frac{1}{32} ln ∣ x + 2 ∣

\int \frac{1}{32 ( x - 2 )} d x = \frac{1}{32} ln ∣ x - 2 ∣

= \frac{1}{16} arctan (\frac{x}{2}) + \frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{32} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{16} arctan (\frac{x}{2}) + \frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{32} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of 24x^5arccot(2x^3)

\int 24 x^{5} arccot (2 x^{3}) d x

integral of 1/(x^2(x^2+4))

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

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

\int \frac{1}{( 2 - 3 x ) ^{2}} d x

integral of sec(2x+4)tan(2x+4)

\int sec (2 x + 4) tan (2 x + 4) d x

integral of (ln(x+1)-ln(x))/(x(x+1))

\int \frac{ln ( x + 1 ) - ln ( x )}{x ( x + 1 )} d x