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

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

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

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

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

Solution steps

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

Simplify

sin (2 x) cos (2 x) tan (2 x) : sin^{2} (2 x)

= \int sin^{2} (2 x) d x

Rewrite using trig identities

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d x = x

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

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

Add a constant to the solution

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

2 sin^{3} (t)

y^{^{''}} - 5 y^{^{'}} + 5 y = 0

\frac{d}{d x} (\frac{3 x + 2}{x ^{2} - 5 x + 3})

\frac{d}{d x} (cos ((8 x^{2} + 9)^{5}))

f (x) g (x)

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