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

The integral of sin(2x)cos(2x) is -1/8 cos(4x)+C

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integral of sin(2x)cos(2x)

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

- \frac{1}{8} cos (4 x) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

= \frac{1}{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

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

Use the common integral:

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

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

Substitute back

u = 4 x

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

Simplify

\frac{1}{2} \cdot \frac{1}{4} (- cos (4 x)) : - \frac{1}{8} cos (4 x)

= - \frac{1}{8} cos (4 x)

Add a constant to the solution

= - \frac{1}{8} cos (4 x) + C

x \to 4 lim (tan (x))

t an g e n t f (x) = 2 x, (32, 8)

d er i v a t i v e f (x) = 3 sin (x) + ln (2 x)

\frac{d}{d x} (3 arcsin (x) + 3 x arcsech (x))

x \to 0 lim (5 - x^{2})

What is the integral of sin(2x)cos(2x) ?
The integral of sin(2x)cos(2x) is -1/8 cos(4x)+C