What is the integral of (sin(2x)) ?

The integral of (sin(2x)) is -1/2 cos(2x)+C

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integral of (sin(2x))

\int (sin (2 x)) d x

- \frac{1}{2} cos (2 x) + C

Solution steps

\int sin (2 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{2} d u

= \int sin (u) \frac{1}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2} (- cos (u))

Substitute back

u = 2 x

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

Simplify

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

Add a constant to the solution

= - \frac{1}{2} cos (2 x) + C

\frac{d y}{d x} = x^{2} - \frac{1}{2} y

\frac{\partial}{\partial x} (x^{3} + 2)

\frac{d}{d x} (x^{4} e^{- 3 x})

x^{2} 8 x - 7

(- 8, - 3), (- 24, 1)