integral from-pi to pi of xsin(ax)

Lösung

\int_{- π}^{π} x sin (a x) d x

- \frac{2 π cos ( πa )}{a} + \frac{2 sin ( πa )}{a ^{2}}

Schritte zur Lösung

\int_{- π}^{π} x sin (a x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Berechne die Grenzen:

- 2 \cdot \frac{π}{a} cos (πa) + \frac{2}{a ^{2}} sin (πa)

= - 2 \cdot \frac{π}{a} cos (πa) + \frac{2}{a ^{2}} sin (πa)

Vereinfache

= - \frac{2 π cos ( πa )}{a} + \frac{2 sin ( πa )}{a ^{2}}

\int_{0}^{1} (x + \frac{1}{2})^{2} d x

\int_{2}^{6} (x - 3) d x

\int_{1}^{7} \frac{4}{x} - \frac{4}{x ^{2}} d x

\int_{0}^{20} π (x)^{2} d x

\int_{2}^{0} x^{4} - 2 x^{3} d x