integral from-pi to 0 of (t+pi/2)cos(nt)

Lösung

\int_{- π}^{0} (t + \frac{π}{2}) cos (n t) d t

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

Schritte zur Lösung

\int_{- π}^{0} (t + \frac{π}{2}) cos (n t) d t

Multipliziere aus

(t + \frac{π}{2}) cos (n t) : t cos (n t) + \frac{π}{2} cos (n t)

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Vereinfache

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

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