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

The integral from-pi to pi of cos(ax) is (2sin(pia))/a

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integral from-pi to pi of cos(ax)

\int_{- π}^{π} cos (a x) d x

\frac{2 sin ( πa )}{a}

Solution steps

\int_{- π}^{π} cos (a x) d x

Apply u-substitution

: \int_{- πa}^{aπ} \frac{cos ( u )}{a} d u

= \int_{- πa}^{aπ} \frac{cos ( u )}{a} d u

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{a} [sin (u)]_{- πa}^{aπ}

Compute the boundaries:

2 sin (πa)

= \frac{1}{a} \cdot 2 sin (πa)

Simplify

= \frac{2 sin ( πa )}{a}

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What is the integral from-pi to pi of cos(ax) ?
The integral from-pi to pi of cos(ax) is (2sin(pia))/a