integral from 0 to 1 of pi(-ln(x))^2

Lösung

\int_{0}^{1} π (- ln (x))^{2} d x

π 2

Dezimale

6.28318 \dots

Schritte zur Lösung

\int_{0}^{1} π (- ln (x))^{2} d x

Entferne die Konstante:

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

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

Vereinfache

(- ln (x))^{2} : ln^{2} (x)

= π \cdot \int_{0}^{1} ln^{2} (x) d x

Wende die partielle Integration an

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

\int 2 ln (x) d x = 2 (x ln (x) - x)

= π [x ln^{2} (x) - 2 (x ln (x) - x)]_{0}^{1}

Berechne die Grenzen:

2

= π 2