What is the integral of sin(2/3 x) ?

The integral of sin(2/3 x) is -3/2 cos((2x)/3)+C

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integral of sin(2/3 x)

\int sin (\frac{2}{3} x) d x

- \frac{3}{2} cos (\frac{2 x}{3}) + C

Solution steps

\int sin (\frac{2}{3} x) d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int sin (\frac{2 x}{3}) d x

Apply u-substitution

: \int sin (u) \frac{3}{2} d u

= \int sin (u) \frac{3}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{3}{2} (- cos (u))

Substitute back

u = \frac{2 x}{3}

= \frac{3}{2} (- cos (\frac{2 x}{3}))

Simplify

= - \frac{3}{2} cos (\frac{2 x}{3})

Add a constant to the solution

= - \frac{3}{2} cos (\frac{2 x}{3}) + C

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What is the integral of sin(2/3 x) ?
The integral of sin(2/3 x) is -3/2 cos((2x)/3)+C