What is the integral from (2pi)/3 to pi of cos(x/2) ?

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

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integral from (2pi)/3 to pi of cos(x/2)

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

2 - 3

Decimal Notation

0.26794 \dots

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Compute the boundaries:

1 - \frac{3}{2}

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

Simplify

= 2 - 3

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