integral from 0 to pi/2 of sin(x^{1/2})

Lösung

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

2 (sin ((\frac{π}{2})^{\frac{1}{2}}) - (\frac{π}{2})^{\frac{1}{2}} cos ((\frac{π}{2})^{\frac{1}{2}}))

Dezimale

1.11754 \dots

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{0}^{(\frac{π}{2})^{\frac{1}{2}}} sin (u) \cdot 2 u d u

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{(\frac{π}{2})^{\frac{1}{2}}} sin (u) u d u

Wende die partielle Integration an

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

\int - cos (u) d u = - sin (u)

= 2 [- u cos (u) - (- sin (u))]_{0}^{(\frac{π}{2})^{\frac{1}{2}}}

Vereinfache

= 2 [- u cos (u) + sin (u)]_{0}^{(\frac{π}{2})^{\frac{1}{2}}}

Berechne die Grenzen:

sin ((\frac{π}{2})^{\frac{1}{2}}) - (\frac{π}{2})^{\frac{1}{2}} cos ((\frac{π}{2})^{\frac{1}{2}})

= 2 (sin ((\frac{π}{2})^{\frac{1}{2}}) - (\frac{π}{2})^{\frac{1}{2}} cos ((\frac{π}{2})^{\frac{1}{2}}))

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