What is the integral from 0 to pi of 1/n cos(nx) ?

The integral from 0 to pi of 1/n cos(nx) is 0

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integral from 0 to pi of 1/n cos(nx)

\int_{0}^{π} \frac{1}{n} cos (n x) d x

0

Solution steps

\int_{0}^{π} \frac{1}{n} cos (n x) d x

Take the constant out:

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

= \frac{1}{n} \cdot \int_{0}^{π} cos (n x) d x

Apply u-substitution

: \int_{0}^{nπ} \frac{cos ( u )}{n} d u

= \frac{1}{n} \cdot \int_{0}^{nπ} \frac{cos ( u )}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \frac{1}{n} \cdot \int_{0}^{nπ} cos (u) d u

Use the common integral:

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

= \frac{1}{n} \cdot \frac{1}{n} [sin (u)]_{0}^{nπ}

Simplify

\frac{1}{n} \cdot \frac{1}{n} [sin (u)]_{0}^{nπ} : \frac{1}{n ^{2}} [sin (u)]_{0}^{πn}

= \frac{1}{n ^{2}} [sin (u)]_{0}^{πn}

Compute the boundaries:

0

= \frac{1}{n ^{2}} \cdot 0

Simplify

= 0

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What is the integral from 0 to pi of 1/n cos(nx) ?
The integral from 0 to pi of 1/n cos(nx) is 0