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

Lösung

integral

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2}}{( x - 2 ) ( x ^{2} + 1 )} : \frac{4}{5 ( x - 2 )} + \frac{x + 2}{5 ( x ^{2} + 1 )}

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

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 1 + y^{2}

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in t e g r a l - 6 x^{3} + 9 x^{2} + 4 x - 3