integral from 0 to 1 of 7x(x^2-1)^5

Lösung

\int_{0}^{1} 7 x (x^{2} - 1)^{5} d x

- \frac{7}{12}

Dezimale

- 0.58333 \dots

Schritte zur Lösung

\int_{0}^{1} 7 x (x^{2} - 1)^{5} d x

Entferne die Konstante:

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

= 7 \cdot \int_{0}^{1} x (x^{2} - 1)^{5} d x

Wende U-Substitution an

= 7 \cdot \int_{- 1}^{0} \frac{u ^{5}}{2} d u

Entferne die Konstante:

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

= 7 \cdot \frac{1}{2} \cdot \int_{- 1}^{0} u^{5} d u

Wende die Potenzregel an

= 7 \cdot \frac{1}{2} [\frac{u ^{6}}{6}]_{- 1}^{0}

Vereinfache

7 \cdot \frac{1}{2} [\frac{u ^{6}}{6}]_{- 1}^{0} : \frac{7}{2} [\frac{u ^{6}}{6}]_{- 1}^{0}

= \frac{7}{2} [\frac{u ^{6}}{6}]_{- 1}^{0}

Berechne die Grenzen:

- \frac{1}{6}

= \frac{7}{2} (- \frac{1}{6})

Vereinfache

= - \frac{7}{12}

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