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

The integral from-pi to pi of cos(kx) is 0

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integral from-pi to pi of cos(kx)

\int_{- π}^{π} cos (k x) d x

0

Solution steps

\int_{- π}^{π} cos (k x) d x

Apply u-substitution

: \int_{- πk}^{kπ} \frac{cos ( u )}{k} d u

= \int_{- πk}^{kπ} \frac{cos ( u )}{k} d u

Take the constant out:

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

= \frac{1}{k} \cdot \int_{- πk}^{kπ} cos (u) d u

Use the common integral:

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

= \frac{1}{k} [sin (u)]_{- πk}^{kπ}

Compute the boundaries:

0

= \frac{1}{k} \cdot 0

Simplify

= 0

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What is the integral from-pi to pi of cos(kx) ?
The integral from-pi to pi of cos(kx) is 0