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

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

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integral of (sin(4x))

\int (sin (4 x)) d x

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

Solution steps

\int sin (4 x) d x

Apply u-substitution

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

= \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

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

Use the common integral:

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

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

Substitute back

u = 4 x

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

Simplify

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

Add a constant to the solution

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

x \to 4 lim (\frac{1}{4 - x})

\frac{d}{d x} (- sin (x^{2}) \cdot 2 x)

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\int (sec^{2} (3 x)) d x

\int_{0}^{r} (r^{2} - y^{2}) d y