What is the integral from-pi to pi of cos(2nx) ?

The integral from-pi to pi of cos(2nx) is (sin(2pin))/n

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integral from-pi to pi of cos(2nx)

\int_{- π}^{π} cos (2 n x) d x

\frac{sin ( 2 πn )}{n}

Solution steps

\int_{- π}^{π} cos (2 n x) d x

Apply u-substitution

: \int_{- 2 πn}^{2 nπ} \frac{cos ( u )}{2 n} d u

= \int_{- 2 πn}^{2 nπ} \frac{cos ( u )}{2 n} d u

Take the constant out:

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

= \frac{1}{2 n} \cdot \int_{- 2 πn}^{2 nπ} cos (u) d u

Use the common integral:

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

= \frac{1}{2 n} [sin (u)]_{- 2 πn}^{2 nπ}

Compute the boundaries:

2 sin (2 πn)

= \frac{1}{2 n} \cdot 2 sin (2 πn)

Simplify

= \frac{sin ( 2 πn )}{n}

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What is the integral from-pi to pi of cos(2nx) ?
The integral from-pi to pi of cos(2nx) is (sin(2pin))/n