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

Lösung

\int_{0}^{1} π (arcsin (x))^{2} d x

π (\frac{π ^{2}}{4} - 2)

Dezimale

1.46838 \dots

Schritte zur Lösung

\int_{0}^{1} π (arcsin (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} (arcsin (x))^{2} d x

Wende die partielle Integration an

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

\int \frac{2 x arcsin ( x )}{1 - x ^{2}} d x = 2 (- arcsin (x) 1 - x^{2} + x)

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

Berechne die Grenzen:

\frac{π ^{2}}{4} - 2

= π (\frac{π ^{2}}{4} - 2)