What is the integral from-pi to pi of xsin(ax) ?

The integral from-pi to pi of xsin(ax) is -(2picos(pia))/a+(2sin(pia))/(a^2)

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integral from-pi to pi of xsin(ax)

\int_{- π}^{π} x sin (a x) d x

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

Solution steps

\int_{- π}^{π} x sin (a x) d x

Apply Integration By Parts

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

= [\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}

Simplify

[\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}

Compute the boundaries:

- 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)

Simplify

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

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

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\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

What is the integral from-pi to pi of xsin(ax) ?
The integral from-pi to pi of xsin(ax) is -(2picos(pia))/a+(2sin(pia))/(a^2)