integral of-2sin^2(x) — step-by-step solution

Let's integrate:

\int(- 2 \cdot {\sin (x)}^{2})dx

We reduce the square using the identity

{\sin (x)}^{2} = \frac{1 - \cos (2 x)}{2}

- \int(1 - \cos (2 x))dx

For the math expression:

- \cos (2 x)

, we know the derivative formula:

\frac{d}{d x} (\sin (a)) = \cos (a)

, so we try differentiating with sin

Let's perform the differentiation:

\frac{d}{d x} (\sin (2 x))

Apply the differentiation rule:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

with f =

2 x

\frac{d}{d x} (2 x) \cdot \cos (2 x)

2 \cdot \cos (2 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}{2} \cdot \sin (2 x) + \int(1)dx)

\frac{1}{2} \cdot \sin (2 x) - \int(1)dx

ℹ

\frac{1}{2} \cdot \sin (2 x)

\frac{\sin (2 x)}{2}

‖

- \int(1)dx

- x

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

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

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