Integral of \( \displaystyle \sin{\left(3 x \right)} \cos{\left(3 x \right)} \)

Problem 4.255 · medium

1. \[ \int \sin{\left(3 x \right)} \cos{\left(3 x \right)}\, dx \]
   integralStarting with the given integral.✓ Proved
2. \[ = \int \frac{\sin{\left(6 x \right)}}{2}\, dx \]
   trig-identityUse the double angle identity sin(2u) = 2sin(u)cos(u) with u = 3x.✓ Proved
3. \[ = \frac{\int \sin{\left(6 x \right)}\, dx}{2} \]
   linearityFactor out the constant 1/2.✓ Proved
4. \[ = - \frac{\cos{\left(6 x \right)}}{12} \]
   antiderivativeThe antiderivative of sin(ax) is -cos(ax)/a.✓ Proved