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Beliebt Rechnen >

integral from-pi to 0 of pisin(nx)

Lösung

\int_{- π}^{0} π sin (n x) d x

Lösung

\frac{π ( - 1 + ( - 1 ) ^{n} )}{n}

Schritte zur Lösung

\int_{- π}^{0} π sin (n x) d x

Entferne die Konstante:

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

= π \cdot \int_{- π}^{0} sin (n x) d x

Wende U-Substitution an

= π \cdot \int_{- πn}^{0} \frac{sin ( u )}{n} d u

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Vereinfache

π \frac{1}{n} [- cos (u)]_{- πn}^{0} : \frac{π}{n} [- cos (u)]_{- πn}^{0}

= \frac{π}{n} [- cos (u)]_{- πn}^{0}

Berechne die Grenzen:

- 1 + (- 1)^{n}

= \frac{π}{n} (- 1 + (- 1)^{n})

Vereinfache

= \frac{π ( - 1 + ( - 1 ) ^{n} )}{n}

Beliebte Beispiele

integral from 0 to 1/2 of x/2

\int_{0}^{\frac{1}{2}} \frac{x}{2} d x

integral from 0 to 1 of sqrt(1+1/(x^2))

\int_{0}^{1} 1 + \frac{1}{x ^{2}} d x

integral from 0 to 2 of sec(x)

\int_{0}^{2} sec (x) d x

integral from 0 to 1 of (5x)

\int_{0}^{1} (5 x) d x

integral from-pi to 0 of-1

\int_{- π}^{0} - 1