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Beliebt Rechnen >

integral of (-5)/((x^2+1)(x-2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{1}{( x ^{2} + 1 ) ( x - 2 )} : \frac{- x - 2}{5 ( x ^{2} + 1 )} + \frac{1}{5 ( x - 2 )}

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

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

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

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int \frac{2 x ^{2}}{( 2 x ^{3} - 5 ) ^{3}} d x

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\int (2 + x)^{3} d x

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\int e^{1 - 4 x} d x