integral from 0 to a of sin((npix)/a)

Lösung

\int_{0}^{a} sin (\frac{nπ x}{a}) d x

\frac{a ( - ( - 1 ) ^{n} + 1 )}{πn}

Schritte zur Lösung

\int_{0}^{a} sin (\frac{nπ x}{a}) d x

Wende U-Substitution an

= \int_{0}^{nπa} \frac{sin ( \frac{u}{a} )}{πn} d u

Entferne die Konstante:

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

= \frac{1}{πn} \cdot \int_{0}^{nπa} sin (\frac{u}{a}) d u

Wende U-Substitution an

= \frac{1}{πn} \cdot \int_{0}^{πn} sin (v) a d v

Entferne die Konstante:

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

= \frac{1}{πn} a \cdot \int_{0}^{πn} sin (v) d v

Nutze das gemeinsame Integral :

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

= \frac{1}{πn} a [- cos (v)]_{0}^{πn}

Berechne die Grenzen:

- (- 1)^{n} + 1

= \frac{1}{πn} a (- (- 1)^{n} + 1)

Vereinfache

= \frac{a ( - ( - 1 ) ^{n} + 1 )}{πn}

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