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

Let's compute the integral:

\int(\cos (3 x))dx

We attempt to reverse an integration: starting from

\cos (3 x)

and

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

, we differentiate using sin

Let's differentiate:

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

Apply the differentiation rule:

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

with f =

3 x

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

3 \cdot \cos (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 \sin (3 x)

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

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

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

▷Result: $\frac{\sin (3 x)}{3} + C$ , with C being any constant