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Beliebt Rechnen >

integral from 0 to pi of cos(x)sin(nx)

Lösung

\int_{0}^{π} cos (x) sin (n x) d x

Lösung

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

Schritte zur Lösung

\int_{0}^{π} cos (x) sin (n x) d x

Verwende die folgenden Identitäten:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{π} sin (n x + x) + sin (n x - 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

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

limit as x approaches 3 of x^2+1

x \to 3 lim (x^{2} + 1)

integral of-t/2

\int - \frac{t}{2} d t

integral from 0 to 4 of-x^2+4x

\int_{0}^{4} - x^{2} + 4 x d x

derivative of (7x^2-1/x)

\frac{d}{d x} (\frac{7 x ^{2} - 1}{x})

y^{''}+4y=0,y(0)=1,y^'(0)=4

y^{^{''}} + 4 y = 0, y (0) = 1, y^{^{'}} (0) = 4