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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{8 2} ln x + 2 + \frac{1}{8 2} ln x - 2 - \frac{1}{4 2} arctan (\frac{x}{2}) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{8 2} ln x + 2 + \frac{1}{8 2} ln x - 2 - \frac{1}{4 2} arctan (\frac{x}{2}) + C

Graph

Beliebte Beispiele

integral of y/(ln(y))

\int \frac{y}{ln ( y )} d y

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integral of x^3sqrt(49+x^2)

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integral of ((x+3))/(sqrt(x))

\int \frac{( x + 3 )}{x} d x