What is the antiderivative of y=(sin2x)(cos2x)?
In an antiderivative, the constant C could be omitted depending on your teacher. I would include it but just in mind, C is not necessary in antiderivatives since antiderivatives are slightly different than integrals.
taking a integral of (sin 2x)(cos2x)
let u=sin 2x
du= 2cos 2x dx
the original integral becomes
int u/2 du
= u^2/2 = (sin 2x)^2/2 +C
it could also be written in cosine form if you use the half angle formulas, but sin is way easier
We'll evaluate the indefinite integral of the given function to determine the antiderivative.
We'll re-write the given function using the double angle identity:
y = 2sin 2x*cos 2x/2
y = sin (4x)/2
`int` sin 2x*cos 2x dx = `int` sin (4x) dx/2
Let 4x = t => 4dx = dt => dx = dt/4
`int` sin (4x) dx/2 = `int` sin t dt/8
`int` sin t dt/8 = - cos t/8 + C
`int` sin 2x*cos 2x dx = - cos (4x)/8 + C
The antiderivative of the given function is the original function Y = - cos (4x)/8 + C.