integral of (x^5-x)/(x^8+1)

Lösung

\int \frac{x ^{5} - x}{x ^{8} + 1} d x

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

Schritte zur Lösung

\int \frac{x ^{5} - x}{x ^{8} + 1} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{u ^{2} - 1}{u ^{4} + 1} : \frac{u ^{2}}{u ^{4} + 1} - \frac{1}{u ^{4} + 1}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Setze in

u = x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int cot^{5} (2 x) csc^{4} (2 x) d x

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