integral from 0 to pi of (1-x)cos(nx)

Lösung

\int_{0}^{π} (1 - x) cos (n x) d x

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

Schritte zur Lösung

\int_{0}^{π} (1 - x) cos (n x) d x

Multipliziere aus

(1 - x) cos (n x) : cos (n x) - x cos (n x)

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

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

\int_{0}^{π} cos (n x) d x = 0

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

= 0 - \frac{( - 1 ) ^{n} - 1}{n ^{2}}

Vereinfache

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

\int_{0}^{2 π} 8 sin (x) cos (x) d x

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