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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

\frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{4}{5} ln ∣ x - 2 ∣ - (\frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{1}{5} ln ∣ x - 2 ∣) : \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1

= \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1 + C

Graph

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