integrate sin(3x) — step-by-step solution

Let's compute the integral:

\int(\sin (3 x))dx

For the math expression:

\sin (3 x)

, we know the derivative formula:

{(\cos (a))}^{'} = - \sin (a)

, so we try differentiating with cos

Let's differentiate:

{(\cos (3 x))}^{'}

Apply the differentiation rule:

{(\cos (f))}^{'} = - {(f)}^{'} \cdot \sin (f)

with f =

3 x

- {(3 x)}^{'} \cdot \sin (3 x)

- 3 \cdot \sin (3 x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

\frac{1}{-3} \cdot \cos (3 x)

- \frac{1}{3} \cdot \cos (3 x)

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

Indefinite integrals are defined up to an additive constant C, so this yields:

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

▷Answer: $- \frac{\cos (3 x)}{3} + C$ , with C being any constant