What is the integral of 2sin(4x) ?

The integral of 2sin(4x) is -1/2 cos(4x)+C

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integral of 2sin(4x)

\int 2 sin (4 x) d x

- \frac{1}{2} cos (4 x) + C

Solution steps

\int 2 sin (4 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 (4 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{4} d u

= 2 \cdot \int sin (u) \frac{1}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} (- cos (u))

Substitute back

u = 4 x

= 2 \cdot \frac{1}{4} (- cos (4 x))

Simplify

2 \cdot \frac{1}{4} (- cos (4 x)) : - \frac{1}{2} cos (4 x)

= - \frac{1}{2} cos (4 x)

Add a constant to the solution

= - \frac{1}{2} cos (4 x) + C

f (x) = x - 4

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(2 x y^{2} + 2 y) d x + (2 x^{2} y + 2 x) d y = 0

\frac{\partial}{\partial x} (100 e^{(- x - 2 y)})