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

Lösung

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

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

Schritte zur Lösung

\int_{- π}^{0} (π + x) cos (n x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

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

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

Vereinfache

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

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