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

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

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integral of sin(x/3)

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

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

Solution steps

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

Apply u-substitution

: \int sin (u) \cdot 3 d u

= \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

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

Use the common integral:

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

= 3 (- cos (u))

Substitute back

u = \frac{x}{3}

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

Simplify

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

Add a constant to the solution

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

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

\frac{\partial}{\partial y} (3 x - 2 y^{4})

\int \frac{x ^{2}}{x ^{3} - 2} d x

x = - 2, x = 3, y = x^{2} - 12, y = x - 6