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

Let's compute the integral:

\int \cos (3 x) d x

To use the known integral of

\cos (u)

, let

u = 3 x

. Then

du = 3 dx

; therefore,

dx = \frac{1}{3} \cdot du

We use

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

, then substitute

u = 3 x

back

ⓘ Differentiating

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

gives

\cos (3 x)

, so

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

is an antiderivative of

\cos (3 x)

\int \cos (3 x) d x

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

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

▶An indefinite integral is defined up to an additive constant C; we therefore obtain:

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