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Beliebt Rechnen >

integral of 1/((x-8)(x^2+25))

Lösung

\int \frac{1}{( x - 8 ) ( x ^{2} + 25 )} d x

Lösung

\frac{1}{89} ln ∣ x - 8 ∣ + \frac{1}{89} (- \frac{1}{2} ln x^{2} + 25 - \frac{8}{5} arctan (\frac{x}{5})) + C

Schritte zur Lösung

\int \frac{1}{( x - 8 ) ( x ^{2} + 25 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 8 ) ( x ^{2} + 25 )} : \frac{1}{89 ( x - 8 )} + \frac{- x - 8}{89 ( x ^{2} + 25 )}

= \int \frac{1}{89 ( x - 8 )} + \frac{- x - 8}{89 ( x ^{2} + 25 )} 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}{89 ( x - 8 )} d x + \int \frac{- x - 8}{89 ( x ^{2} + 25 )} d x

\int \frac{1}{89 ( x - 8 )} d x = \frac{1}{89} ln ∣ x - 8 ∣

\int \frac{- x - 8}{89 ( x ^{2} + 25 )} d x = \frac{1}{89} (- \frac{1}{2} ln x^{2} + 25 - \frac{8}{5} arctan (\frac{x}{5}))

= \frac{1}{89} ln ∣ x - 8 ∣ + \frac{1}{89} (- \frac{1}{2} ln x^{2} + 25 - \frac{8}{5} arctan (\frac{x}{5}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{89} ln ∣ x - 8 ∣ + \frac{1}{89} (- \frac{1}{2} ln x^{2} + 25 - \frac{8}{5} arctan (\frac{x}{5})) + C

Graph

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