integral of sin(-4x) — step-by-step solution

Before performing the integration, let's calculate:

\sin (-4 x)

ⓘ sin is odd, sin(-x) = -sin(x):

\sin (- 4 x)

- \sin (4 x)

- \sin (4 x)

Let's compute the integral:

\int (- \sin (4 x)) d x

To use the known integral of

\sin (u)

, let

u = 4 x

. Then

du = 4 dx

, so

dx = \frac{1}{4} \cdot du

We use

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

, then substitute

u = 4 x

back

ⓘ The derivative of

- (\frac{- \cos (4 x)}{4})

- \sin (4 x)

, so

- (\frac{- \cos (4 x)}{4})

is an antiderivative of

- \sin (4 x)

\int (- \sin (4 x)) d x

- (\frac{- \cos (4 x)}{4})

- (\frac{- \cos (4 x)}{4})

ⓘ Two minus signs cancel out:

- (\frac{- \cos (4 x)}{4})

\frac{\cos (4 x)}{4}

\frac{\cos (4 x)}{4}

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

\frac{\cos (4 x)}{4} + C