integral from 1 to 4 of 3/2 sin(2-x/2)

Lösung

\int_{1}^{4} \frac{3}{2} sin (2 - \frac{x}{2}) d x

3 cos^{2} (2) - 3 cos (2) cos (\frac{1}{2}) + 3 sin^{2} (2) - 3 sin (2) sin (\frac{1}{2})

Dezimale

2.78778 \dots

Schritte zur Lösung

\int_{1}^{4} \frac{3}{2} sin (2 - \frac{x}{2}) d x

Entferne die Konstante:

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

= \frac{3}{2} \cdot \int_{1}^{4} sin (2 - \frac{x}{2}) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{3}{2} \cdot \int_{1}^{4} - cos (2) sin (\frac{x}{2}) + cos (\frac{x}{2}) sin (2) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{2} (- \int_{1}^{4} cos (2) sin (\frac{x}{2}) d x + \int_{1}^{4} cos (\frac{x}{2}) sin (2) d x)

\int_{1}^{4} cos (2) sin (\frac{x}{2}) d x = - 2 cos^{2} (2) + 2 cos (2) cos (\frac{1}{2})

\int_{1}^{4} cos (\frac{x}{2}) sin (2) d x = 2 sin^{2} (2) - 2 sin (2) sin (\frac{1}{2})

= \frac{3}{2} (- (- 2 cos^{2} (2) + 2 cos (2) cos (\frac{1}{2})) + 2 sin^{2} (2) - 2 sin (2) sin (\frac{1}{2}))

Vereinfache

\frac{3}{2} (- (- 2 cos^{2} (2) + 2 cos (2) cos (\frac{1}{2})) + 2 sin^{2} (2) - 2 sin (2) sin (\frac{1}{2})) : 3 cos^{2} (2) - 3 cos (2) cos (\frac{1}{2}) + 3 sin^{2} (2) - 3 sin (2) sin (\frac{1}{2})

= 3 cos^{2} (2) - 3 cos (2) cos (\frac{1}{2}) + 3 sin^{2} (2) - 3 sin (2) sin (\frac{1}{2})

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