What is the integral from pi/6 to pi/3 of cos(3x) ?

The integral from pi/6 to pi/3 of cos(3x) is -1/3

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integral from pi/6 to pi/3 of cos(3x)

\int_{\frac{π}{6}}^{\frac{π}{3}} cos (3 x) d x

- \frac{1}{3}

Decimal Notation

- 0.33333 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{3}} cos (3 x) d x

Apply u-substitution

: \int_{\frac{π}{2}}^{π} cos (u) \frac{1}{3} d u

= \int_{\frac{π}{2}}^{π} cos (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{\frac{π}{2}}^{π} cos (u) d u

Use the common integral:

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

= \frac{1}{3} [sin (u)]_{\frac{π}{2}}^{π}

Compute the boundaries:

- 1

= \frac{1}{3} (- 1)

Simplify

= - \frac{1}{3}

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What is the integral from pi/6 to pi/3 of cos(3x) ?
The integral from pi/6 to pi/3 of cos(3x) is -1/3