integral from 0 to pi/4 of cos(sqrt(x))

Lösung

\int_{0}^{\frac{π}{4}} cos (x) d x

π sin (\frac{π}{2}) + 2 cos (\frac{π}{2}) - 2

Dezimale

0.63778 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{4}} cos (x) d x

Wende U-Substitution an

= \int_{0}^{\frac{π}{2}} cos (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}} cos (u) u d u

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

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

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

Vereinfache

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

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