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Beliebt Rechnen >

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

Lösung

\int_{0}^{π} (π - x) cos (n x) d x

Lösung

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

Schritte zur Lösung

\int_{0}^{π} (π - x) cos (n x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Berechne die Grenzen:

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

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

Vereinfache

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

Beliebte Beispiele

limit as x approaches 6 of sqrt(x-10)

x \to 6 lim (x - 10)

y^{''}+5y^'-6y=0

y^{^{''}} + 5 y^{^{'}} - 6 y = 0

derivative y=ln(2+x)+2/(2+x)

d er i v a t i v e y = ln (2 + x) + \frac{2}{2 + x}

derivative sqrt(6x-1)

d er i v a t i v e 6 x - 1

(\partial)/(\partial x)(sin(x)+pi*cos(y))

\frac{\partial}{\partial x} (sin (x) + π \cdot cos (y))