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

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

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integral of (2sin(2x))

\int (2 sin (2 x)) d x

- cos (2 x) + C

Solution steps

\int 2 sin (2 x) d x

Take the constant out:

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

= 2 \cdot \int sin (2 x) d x

Apply u-substitution

= 2 \cdot \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

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

Use the common integral:

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

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

Substitute back

u = 2 x

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

Simplify

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

= - cos (2 x)

Add a constant to the solution

= - cos (2 x) + C