integral from 0 to 1/2 of xarcsin(x)

Lösung

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

\frac{- π + 3 3}{48}

Dezimale

0.04280 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{x ^{2}}{2 1 - x ^{2}} d x = \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

= [\frac{1}{2} (x^{2} arcsin (x) - 2 \cdot \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))))]_{0}^{\frac{1}{2}}

2 \cdot \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) = \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

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

Berechne die Grenzen:

\frac{- π + 3 3}{48}

= \frac{- π + 3 3}{48}

\int_{0}^{2 π} (sin (x))^{2} d x

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (x) d x

\int_{0}^{\frac{π}{2}} cos (θ) sin (θ) d θ

\int_{- 1}^{1} x cos (x) d x

\int_{0}^{π} x^{2} d x