integral from 2 to 4 of pi(-x^2+6x-8)^2

Lösung

\int_{2}^{4} π (- x^{2} + 6 x - 8)^{2} d x

π \frac{16}{15}

Dezimale

3.35103 \dots

Schritte zur Lösung

\int_{2}^{4} π (- x^{2} + 6 x - 8)^{2} d x

Entferne die Konstante:

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

= π \cdot \int_{2}^{4} (- x^{2} + 6 x - 8)^{2} d x

Multipliziere aus

(- x^{2} + 6 x - 8)^{2} : x^{4} - 12 x^{3} + 52 x^{2} - 96 x + 64

= π \cdot \int_{2}^{4} x^{4} - 12 x^{3} + 52 x^{2} - 96 x + 64 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_{2}^{4} x^{4} d x - \int_{2}^{4} 12 x^{3} d x + \int_{2}^{4} 52 x^{2} d x - \int_{2}^{4} 96 x d x + \int_{2}^{4} 64 d x)

\int_{2}^{4} x^{4} d x = \frac{992}{5}

\int_{2}^{4} 12 x^{3} d x = 720

\int_{2}^{4} 52 x^{2} d x = \frac{2912}{3}

\int_{2}^{4} 96 x d x = 576

\int_{2}^{4} 64 d x = 128

= π (\frac{992}{5} - 720 + \frac{2912}{3} - 576 + 128)

Vereinfache

π (\frac{992}{5} - 720 + \frac{2912}{3} - 576 + 128) : π \frac{16}{15}

= π \frac{16}{15}

\int_{1}^{4} 2^{5 x} d x

\int_{0}^{\infty} x^{4} d x

\int_{2}^{\infty} - 9 x^{- 2} d x

\int_{1}^{6} (\frac{5}{x}) d x

\int_{1}^{3} x^{2} y d y