What is the integral of-3cos(x/3) ?

The integral of-3cos(x/3) is -9sin(x/3)+C

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integral of-3cos(x/3)

\int - 3 cos (\frac{x}{3}) d x

- 9 sin (\frac{x}{3}) + C

Solution steps

\int - 3 cos (\frac{x}{3}) d x

Take the constant out:

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

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

Apply u-substitution

= - 3 \cdot \int cos (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 3 \cdot \int cos (u) d u

Use the common integral:

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

= - 3 \cdot 3 sin (u)

Substitute back

u = \frac{x}{3}

= - 3 \cdot 3 sin (\frac{x}{3})

Simplify

= - 9 sin (\frac{x}{3})

Add a constant to the solution

= - 9 sin (\frac{x}{3}) + C