integral of x/((x+10)(x^2+12))

Lösung

\int \frac{x}{( x + 10 ) ( x ^{2} + 12 )} d x

- \frac{5}{56} ln ∣ x + 10 ∣ + \frac{1}{56} (\frac{5}{2} ln x^{2} + 12 + 3 arctan (\frac{x}{2 3})) + C

Schritte zur Lösung

\int \frac{x}{( x + 10 ) ( x ^{2} + 12 )} d x

Ermittle den Partialbruch von

\frac{x}{( x + 10 ) ( x ^{2} + 12 )} : - \frac{5}{56 ( x + 10 )} + \frac{5 x + 6}{56 ( x ^{2} + 12 )}

= \int - \frac{5}{56 ( x + 10 )} + \frac{5 x + 6}{56 ( x ^{2} + 12 )} 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{5}{56 ( x + 10 )} d x + \int \frac{5 x + 6}{56 ( x ^{2} + 12 )} d x

\int \frac{5}{56 ( x + 10 )} d x = \frac{5}{56} ln ∣ x + 10 ∣

\int \frac{5 x + 6}{56 ( x ^{2} + 12 )} d x = \frac{1}{56} (\frac{5}{2} ln x^{2} + 12 + 3 arctan (\frac{x}{2 3}))

= - \frac{5}{56} ln ∣ x + 10 ∣ + \frac{1}{56} (\frac{5}{2} ln x^{2} + 12 + 3 arctan (\frac{x}{2 3}))

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{56} ln ∣ x + 10 ∣ + \frac{1}{56} (\frac{5}{2} ln x^{2} + 12 + 3 arctan (\frac{x}{2 3})) + C

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