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Beliebt Rechnen >

integral of (x^2-12x-1)/((x-1)^2(x^2+1))

Lösung

\int \frac{x ^{2} - 12 x - 1}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

Lösung

ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{1}{2} ln x^{2} + 1 + 6 arctan (x) + C

Schritte zur Lösung

\int \frac{x ^{2} - 12 x - 1}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

\frac{x ^{2} - 12 x - 1}{( x - 1 ) ^{2} ( x ^{2} + 1 )} : \frac{1}{x - 1} - \frac{6}{( x - 1 ) ^{2}} + \frac{- x + 6}{x ^{2} + 1}

= \int \frac{1}{x - 1} - \frac{6}{( x - 1 ) ^{2}} + \frac{- x + 6}{x ^{2} + 1} 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}{x - 1} d x - \int \frac{6}{( x - 1 ) ^{2}} d x + \int \frac{- x + 6}{x ^{2} + 1} d x

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

\int \frac{6}{( x - 1 ) ^{2}} d x = - \frac{6}{x - 1}

\int \frac{- x + 6}{x ^{2} + 1} d x = - \frac{1}{2} ln x^{2} + 1 + 6 arctan (x)

= ln ∣ x - 1 ∣ - (- \frac{6}{x - 1}) - \frac{1}{2} ln x^{2} + 1 + 6 arctan (x)

Vereinfache

= ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{1}{2} ln x^{2} + 1 + 6 arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ + \frac{6}{x - 1} - \frac{1}{2} ln x^{2} + 1 + 6 arctan (x) + C

Graph

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