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Beliebt Rechnen >

integral from 0 to 1 of 2pix(15-15x)

Lösung

\int_{0}^{1} 2 π x (15 - 15 x) d x

Lösung

5 π

+1

Dezimale

15.70796 \dots

Schritte zur Lösung

\int_{0}^{1} 2 π x (15 - 15 x) d x

Entferne die Konstante:

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

= 2 π \cdot \int_{0}^{1} x (15 - 15 x) d x

Wende die partielle Integration an

= 2 π [\frac{1}{2} x^{2} (15 - 15 x) - \int - \frac{15 x ^{2}}{2} d x]_{0}^{1}

\int - \frac{15 x ^{2}}{2} d x = - \frac{5 x ^{3}}{2}

= 2 π [\frac{1}{2} x^{2} (15 - 15 x) - (- \frac{5 x ^{3}}{2})]_{0}^{1}

Vereinfache

2 π [\frac{1}{2} x^{2} (15 - 15 x) - (- \frac{5 x ^{3}}{2})]_{0}^{1} : 2 π [\frac{1}{2} (15 x^{2} - 10 x^{3})]_{0}^{1}

= 2 π [\frac{1}{2} (15 x^{2} - 10 x^{3})]_{0}^{1}

Berechne die Grenzen:

\frac{5}{2}

= 2 π \frac{5}{2}

Vereinfache

= 5 π

Graph

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