What is the integral from pi/6 to pi/2 of cos(2x) ?

The integral from pi/6 to pi/2 of cos(2x) is -(sqrt(3))/4

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integral from pi/6 to pi/2 of cos(2x)

\int_{\frac{π}{6}}^{\frac{π}{2}} cos (2 x) d x

- \frac{3}{4}

Decimal

- 0.43301 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{2}} cos (2 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Compute the boundaries:

- \frac{3}{2}

= \frac{1}{2} (- \frac{3}{2})

Simplify

= - \frac{3}{4}

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What is the integral from pi/6 to pi/2 of cos(2x) ?
The integral from pi/6 to pi/2 of cos(2x) is -(sqrt(3))/4