What is the antiderivative of y=(sin2x)(cos2x)?

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shaznl1's profile pic

shaznl1 | High School Teacher | (Level 1) Salutatorian

Posted on

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

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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