What is the integral from 1/2 to 1/3 of (cos(piw)) ?

The integral from 1/2 to 1/3 of (cos(piw)) is -(2-sqrt(3))/(2pi)

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integral from 1/2 to 1/3 of (cos(piw))

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

- \frac{2 - 3}{2 π}

Decimal

- 0.04264 \dots

Solution steps

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Compute the boundaries:

1 - \frac{3}{2}

= - \frac{1}{π} (1 - \frac{3}{2})

Simplify

= - \frac{2 - 3}{2 π}

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What is the integral from 1/2 to 1/3 of (cos(piw)) ?
The integral from 1/2 to 1/3 of (cos(piw)) is -(2-sqrt(3))/(2pi)