Integral of \( \displaystyle \sin{\left(2 x \right)} \cos{\left(2 x \right)} \)

Problem 4.561 · medium

1. \[ \int \sin{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{\sin{\left(4 x \right)}}{2}\, dx \]
   trig-identityUse the double angle identity sin(2u) = 2sin(u)cos(u) with u = 2x.✓ Proved
3. \[ = \frac{\int \sin{\left(4 x \right)}\, dx}{2} \]
   linearityFactor out the constant 1/2.✓ Proved
4. \[ = - \frac{\cos{\left(4 x \right)}}{8} \]
   antiderivativeIntegrate sin(4*x) to get -cos(4*x)/4 and multiply by 1/2.✓ Proved