How calculate integral of y=sin2xcos2x without substitutions?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You may re-write the integrand using the double angle identity, such that:

`sin 2x*cos 2x = (sin 2*(2x))/2`

Hence, you may replace `(sin 4x)/2` for the integrand `sin 2x*cos 2x` , such that:

`int sin 2x*cos 2x dx = int(sin 4x)/2 dx`

Taking out the constant `1/2` , yields:

`int(sin 4x)/2 dx = (1/2) int sin 4x dx`

`int(sin 4x)/2 dx = -(1/2)*(cos 4x)/4 + c`

`int(sin 4x)/2 dx = -(cos 4x)/8 + c`

Hence, evaluating the indefinite integral of the given function, using the double angle trigonometric identity to avoid the substitution method, yields `int sin 2x*cos 2x dx = -(cos 4x)/8 + c.`