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Beliebt Rechnen >

integral of x(1+x^2)^7

Lösung

\int x (1 + x^{2})^{7} d x

Lösung

\frac{1}{16} (1 + x^{2})^{8} + C

Schritte zur Lösung

\int x (1 + x^{2})^{7} d x

Wende U-Substitution an

= \int \frac{u ^{7}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int u^{7} d u

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{u ^{8}}{8}

Setze in

u = 1 + x^{2}

ein

= \frac{1}{2} \cdot \frac{( 1 + x ^{2} ) ^{8}}{8}

Vereinfache

\frac{1}{2} \cdot \frac{( 1 + x ^{2} ) ^{8}}{8} : \frac{1}{16} (1 + x^{2})^{8}

= \frac{1}{16} (1 + x^{2})^{8}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{16} (1 + x^{2})^{8} + C

Graph

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