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Beliebt Rechnen >

integral of 8/((x+2)(x^2+4))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

= 8 \cdot \int \frac{1}{8 ( x + 2 )} + \frac{- x + 2}{8 ( 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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

tangent (-5x^2)/(2x+3)

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\frac{d}{d x} (arcsin (\frac{a}{x}))

integral of (x^3-2x^2+5x-2)/(x^3-2x^2+x)

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\frac{d}{d x} (- 2 sin^{2} (x))

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n = 1 \sum \infty \frac{( ln ( 3 n ) )}{n}