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

The integral of 2ysin(2x) is -ycos(2x)+C

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integral of 2ysin(2x)

\int 2 y sin (2 x) d x

- y cos (2 x) + C

Solution steps

\int 2 y sin (2 x) d x

Take the constant out:

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

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

Apply u-substitution

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

Use the common integral:

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

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

Substitute back

u = 2 x

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

Simplify

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

= - y cos (2 x)

Add a constant to the solution

= - y cos (2 x) + C