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Beliebt Rechnen >

integral of (5x^2)/((x-2)(x^2+4))

Lösung

\int \frac{5 x ^{2}}{( x - 2 ) ( x ^{2} + 4 )} d x

Lösung

5 (\frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}))) + C

Schritte zur Lösung

\int \frac{5 x ^{2}}{( x - 2 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{x ^{2}}{( x - 2 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

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

= 5 \cdot \int \frac{1}{2 ( x - 2 )} + \frac{x + 2}{2 ( x ^{2} + 4 )} 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

= 5 (\int \frac{1}{2 ( x - 2 )} d x + \int \frac{x + 2}{2 ( x ^{2} + 4 )} d x)

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

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

= 5 (\frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2})))

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}))) + C

Graph

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\int tan^{5} (4 x) d x

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