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Beliebt Rechnen >

integral of x^7arctan(x^4)

Lösung

\int x^{7} arctan (x^{4}) d x

Lösung

\frac{1}{4} (\frac{1}{2} x^{8} arctan (x^{4}) - \frac{1}{2} (- arctan (x^{4}) + x^{4})) + C

Schritte zur Lösung

\int x^{7} arctan (x^{4}) d x

Wende U-Substitution an

= \int \frac{u arctan ( u )}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int u arctan (u) d u

Wende die partielle Integration an

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

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

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

Setze in

u = x^{4}

ein

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

(x^{4})^{2} = x^{8}

= \frac{1}{4} (\frac{1}{2} x^{8} arctan (x^{4}) - \frac{1}{2} (- arctan (x^{4}) + x^{4}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (\frac{1}{2} x^{8} arctan (x^{4}) - \frac{1}{2} (- arctan (x^{4}) + x^{4})) + C

Graph

Beliebte Beispiele

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