What is the integral from 0 to pi/3 of 6sin(x/2) ?

The integral from 0 to pi/3 of 6sin(x/2) is 12(-(sqrt(3))/2+1)

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integral from 0 to pi/3 of 6sin(x/2)

\int_{0}^{\frac{π}{3}} 6 sin (\frac{x}{2}) d x

12 (- \frac{3}{2} + 1)

Decimal

1.60769 \dots

Solution steps

\int_{0}^{\frac{π}{3}} 6 sin (\frac{x}{2}) d x

Take the constant out:

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

= 6 \cdot \int_{0}^{\frac{π}{3}} sin (\frac{x}{2}) d x

Apply u-substitution

: \int_{0}^{\frac{π}{6}} sin (u) \cdot 2 d u

= 6 \cdot \int_{0}^{\frac{π}{6}} sin (u) \cdot 2 d u

Take the constant out:

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

= 6 \cdot 2 \cdot \int_{0}^{\frac{π}{6}} sin (u) d u

Use the common integral:

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

= 6 \cdot 2 [- cos (u)]_{0}^{\frac{π}{6}}

Simplify

= 12 [- cos (u)]_{0}^{\frac{π}{6}}

Compute the boundaries:

- \frac{3}{2} + 1

= 12 (- \frac{3}{2} + 1)

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What is the integral from 0 to pi/3 of 6sin(x/2) ?
The integral from 0 to pi/3 of 6sin(x/2) is 12(-(sqrt(3))/2+1)