Final answer:
The student's query includes typos but seems related to trigonometry or kinematics involving waves. It touches upon trigonometric identities such as double angle formulas and the laws of sines and cosines. Unfortunately, without a correct version of the question, it is not possible to give an exact answer.
Step-by-step explanation:
The original question posted by the student seems to have a typo, but based on the provided reference information, it appears that the question may involve the calculation of a trigonometric integral, possibly in the context of applying trigonometric identities or solving a problem involving the kinematics of waves.
It may be helpful to review the trigonometric identities for sum and difference of angles, the double angle formulas for sin and cos, and the laws of sines and cosines which are applicable for solving various problems in trigonometry involving triangles, oscillations, or the calculation of vector magnitudes.
For the specific integral of the form ∑ cos(x) sin(x) dx, recalling that sin(2x) = 2sin(x)cos(x), would be necessary to solve it. However, without additional context or a correct version of the student's original question, providing a definitive answer is not possible.