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

The integral of 2sin(x/3) is -6cos(x/3)+C

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integral of 2sin(x/3)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int sin (u) \cdot 3 d u

Take the constant out:

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

= 2 \cdot 3 \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot 3 (- cos (u))

Substitute back

u = \frac{x}{3}

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

Simplify

= - 6 cos (\frac{x}{3})

Add a constant to the solution

= - 6 cos (\frac{x}{3}) + C