integral from 0 to pi of (pi-x)*cos(nx)

Lösung

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

\frac{- ( - 1 ) ^{n} + 1}{n ^{2}}

Schritte zur Lösung

\int_{0}^{π} (π - x) cos (n x) d x

Wende die partielle Integration an

= [\frac{1}{n} (sin (n x) (π - x) - n \cdot \int - \frac{1}{n} sin (n x) d x)]_{0}^{π}

\int - \frac{1}{n} sin (n x) d x = \frac{1}{n ^{2}} cos (n x)

= [\frac{1}{n} (sin (n x) (π - x) - n \frac{1}{n ^{2}} cos (n x))]_{0}^{π}

Vereinfache

[\frac{1}{n} (sin (n x) (π - x) - n \frac{1}{n ^{2}} cos (n x))]_{0}^{π} : [\frac{1}{n} (sin (n x) (π - x) - \frac{1}{n} cos (n x))]_{0}^{π}

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

Berechne die Grenzen:

- \frac{( - 1 ) ^{n}}{n ^{2}} + \frac{1}{n ^{2}}

= - \frac{( - 1 ) ^{n}}{n ^{2}} + \frac{1}{n ^{2}}

Vereinfache

= \frac{- ( - 1 ) ^{n} + 1}{n ^{2}}

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