integral from 1 to 3 of pi(1/x+1)^2

Lösung

\int_{1}^{3} π (\frac{1}{x} + 1)^{2} d x

π (2 ln (3) + \frac{8}{3})

Dezimale

15.28036 \dots

Schritte zur Lösung

\int_{1}^{3} π (\frac{1}{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_{1}^{3} (\frac{1}{x} + 1)^{2} d x

Multipliziere aus

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

= π \cdot \int_{1}^{3} \frac{1}{x ^{2}} + \frac{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_{1}^{3} \frac{1}{x ^{2}} d x + \int_{1}^{3} \frac{2}{x} d x + \int_{1}^{3} 1 d x)

\int_{1}^{3} \frac{1}{x ^{2}} d x = \frac{2}{3}

\int_{1}^{3} \frac{2}{x} d x = 2 ln (3)

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

= π (\frac{2}{3} + 2 ln (3) + 2)

Ziehe Brüche zusammen

2 + \frac{2}{3} : \frac{8}{3}

= π (2 ln (3) + \frac{8}{3})

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