integral of (2-x)sin((npix)/2)

Lösung

\int (2 - x) sin (\frac{nπ x}{2}) d x

- \frac{4}{πn} cos (\frac{πn}{2} x) - \frac{4}{π ^{2} n ^{2}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2})) + C

Schritte zur Lösung

\int (2 - x) sin (\frac{nπ x}{2}) d x

Multipliziere aus

(2 - x) sin (\frac{nπ x}{2}) : 2 sin (\frac{πn x}{2}) - x sin (\frac{πn x}{2})

= \int 2 sin (\frac{πn x}{2}) - x sin (\frac{πn x}{2}) 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 2 sin (\frac{πn x}{2}) d x - \int x sin (\frac{πn x}{2}) d x

\int 2 sin (\frac{πn x}{2}) d x = - \frac{4}{πn} cos (\frac{πn}{2} x)

\int x sin (\frac{πn x}{2}) d x = \frac{4}{π ^{2} n ^{2}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))

= - \frac{4}{πn} cos (\frac{πn}{2} x) - \frac{4}{π ^{2} n ^{2}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2}))

Füge eine Konstante zur Lösung hinzu

= - \frac{4}{πn} cos (\frac{πn}{2} x) - \frac{4}{π ^{2} n ^{2}} (- \frac{πn}{2} x cos (\frac{πn x}{2}) + sin (\frac{πn x}{2})) + C

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