integral from 0 to 1 of sin^2(npix)

Lösung

\int_{0}^{1} sin^{2} (nπ x) d x

\frac{1}{2} - \frac{1}{4 πn} sin (2 πn)

Schritte zur Lösung

\int_{0}^{1} sin^{2} (nπ x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{1} \frac{1 - cos ( 2 nπ x )}{2} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{1} 1 - cos (2 nπ x) d x

Wende U-Substitution an

= \frac{1}{2} \cdot \int_{0}^{2 πn} \frac{1 - cos ( u )}{2 πn} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2 πn} \cdot \int_{0}^{2 πn} 1 - cos (u) d u

Wende die Summenregel an:

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

= \frac{1}{2} \cdot \frac{1}{2 πn} (\int_{0}^{2 πn} 1 d u - \int_{0}^{2 πn} cos (u) d u)

\int_{0}^{2 πn} 1 d u = 2 πn

\int_{0}^{2 πn} cos (u) d u = sin (2 πn)

= \frac{1}{2} \cdot \frac{1}{2 πn} (2 πn - sin (2 πn))

Vereinfache

\frac{1}{2} \cdot \frac{1}{2 πn} (2 πn - sin (2 πn)) : \frac{1}{2} - \frac{1}{4 πn} sin (2 πn)

= \frac{1}{2} - \frac{1}{4 πn} sin (2 πn)

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