integral from pi/2 to pi of xsin(nx)

Lösung

\int_{\frac{π}{2}}^{π} x sin (n x) d x

- \frac{π ( - 1 ) ^{n}}{n} - \frac{- πn cos ( \frac{πn}{2} ) + 2 sin ( \frac{πn}{2} )}{2 n ^{2}}

Schritte zur Lösung

\int_{\frac{π}{2}}^{π} x sin (n x) d x

Wende die partielle Integration an

= [\frac{1}{n} (- x cos (n x) - n \cdot \int - \frac{1}{n} cos (n x) d x)]_{\frac{π}{2}}^{π}

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

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

Vereinfache

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

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

Berechne die Grenzen:

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

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

Vereinfache

= - \frac{π ( - 1 ) ^{n}}{n} - \frac{- πn cos ( \frac{πn}{2} ) + 2 sin ( \frac{πn}{2} )}{2 n ^{2}}

\int_{3}^{4} - 6 x d x

\int_{- 1}^{5} 2 x d x

\int_{0}^{1} (7 x^{e} + 4 e^{x}) d x

\int_{0}^{0.43} 75.7575 x d x

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