integral from 0 to 7 of pi(x+1)^2

Lösung

\int_{0}^{7} π (x + 1)^{2} d x

π \frac{511}{3}

Dezimale

535.11794 \dots

Schritte zur Lösung

\int_{0}^{7} π (x + 1)^{2} d x

Entferne die Konstante:

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

= π \cdot \int_{0}^{7} (x + 1)^{2} d x

Multipliziere aus

(x + 1)^{2} : x^{2} + 2 x + 1

= π \cdot \int_{0}^{7} x^{2} + 2 x + 1 d x

Wende die Summenregel an:

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

= π (\int_{0}^{7} x^{2} d x + \int_{0}^{7} 2 x d x + \int_{0}^{7} 1 d x)

\int_{0}^{7} x^{2} d x = \frac{343}{3}

\int_{0}^{7} 2 x d x = 49

\int_{0}^{7} 1 d x = 7

= π (\frac{343}{3} + 49 + 7)

Vereinfache

π (\frac{343}{3} + 49 + 7) : π \frac{511}{3}

= π \frac{511}{3}

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