What is the integral from 0 to pi of xcos(ax) ?

The integral from 0 to pi of xcos(ax) is (piasin(pia)+cos(pia)-1)/(a^2)

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integral from 0 to pi of xcos(ax)

\int_{0}^{π} x cos (a x) d x

\frac{πa sin ( πa ) + cos ( πa ) - 1}{a ^{2}}

Solution steps

\int_{0}^{π} x cos (a x) d x

Apply u-substitution

= \int_{0}^{aπ} \frac{u cos ( u )}{a ^{2}} d u

Take the constant out:

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

= \frac{1}{a ^{2}} \cdot \int_{0}^{aπ} u cos (u) d u

Apply Integration By Parts

= \frac{1}{a ^{2}} [u sin (u) - \int sin (u) d u]_{0}^{πa}

\int sin (u) d u = - cos (u)

= \frac{1}{a ^{2}} [u sin (u) - (- cos (u))]_{0}^{πa}

Simplify

= \frac{1}{a ^{2}} [u sin (u) + cos (u)]_{0}^{πa}

Compute the boundaries:

πa sin (πa) + cos (πa) - 1

= \frac{1}{a ^{2}} (πa sin (πa) + cos (πa) - 1)

Simplify

= \frac{πa sin ( πa ) + cos ( πa ) - 1}{a ^{2}}

What is the integral from 0 to pi of xcos(ax) ?
The integral from 0 to pi of xcos(ax) is (piasin(pia)+cos(pia)-1)/(a^2)