integral of (x^6)/(x^4+1)

Lösung

\int \frac{x ^{6}}{x ^{4} + 1} d x

\frac{x ^{3}}{3} + \frac{1}{4 2} ln 2 x^{2} + 22 x + 2 - \frac{1}{2 2} arctan (2 x + 1) - \frac{1}{4 2} ln 2 x^{2} - 22 x + 2 - \frac{1}{2 2} arctan (2 x - 1) + C

Schritte zur Lösung

\int \frac{x ^{6}}{x ^{4} + 1} d x

Lange Division

\frac{x ^{6}}{x ^{4} + 1} : x^{2} - \frac{x ^{2}}{x ^{4} + 1}

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{x ^{2}}{x ^{4} + 1} d x = - \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))

= \frac{x ^{3}}{3} - (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

Vereinfache

\frac{x ^{3}}{3} - (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) : \frac{x ^{3}}{3} + \frac{1}{4 2} ln 2 x^{2} + 22 x + 2 - \frac{1}{2 2} arctan (2 x + 1) - \frac{1}{4 2} ln 2 x^{2} - 22 x + 2 - \frac{1}{2 2} arctan (2 x - 1)

= \frac{x ^{3}}{3} + \frac{1}{4 2} ln 2 x^{2} + 22 x + 2 - \frac{1}{2 2} arctan (2 x + 1) - \frac{1}{4 2} ln 2 x^{2} - 22 x + 2 - \frac{1}{2 2} arctan (2 x - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{1}{4 2} ln 2 x^{2} + 22 x + 2 - \frac{1}{2 2} arctan (2 x + 1) - \frac{1}{4 2} ln 2 x^{2} - 22 x + 2 - \frac{1}{2 2} arctan (2 x - 1) + C

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