integrate sin(2x) — step-by-step solution

Let's integrate:

\int(\sin (2 x))dx

We attempt to reverse an integration: starting from

\sin (2 x)

and

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

, we differentiate using cos

Let's differentiate:

{(\cos (2 x))}^{'}

We differentiate using:

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

(where f =

2 x

)

- {(2 x)}^{'} \cdot \sin (2 x)

- 2 \cdot \sin (2 x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

\frac{1}{-2} \cdot \cos (2 x)

- \frac{1}{2} \cdot \cos (2 x)

- \frac{\cos (2 x)}{2}

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

- \frac{\cos (2 x)}{2} + C

▷Answer: $- \frac{\cos (2 x)}{2} + C$ , with C being any constant