integral from-pi to 0 of (pi+t)cos(nt)

Lösung

\int_{- π}^{0} (π + t) cos (n t) d t

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

Schritte zur Lösung

\int_{- π}^{0} (π + t) cos (n t) d t

Wende die partielle Integration an

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

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

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

Vereinfache

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