Final answer:
The integral ∫ √(√2/3) cos(2x) dx is solved by factoring out the constant and then applying integration by parts, resulting in a final evaluated form.
Step-by-step explanation:
To evaluate the integral ∫ √(√2/3) cos(2x) dx, we first look for a substitution that simplifies the integral. However, in this case, the constant √(√2/3) can be factored out of the integral, leaving us with √(√2/3) ∫ cos(2x) dx. Next, we use integration by parts, which is based on the formula ∫ u dv = uv - ∫ v du. Letting u = cos(2x) and dv = dx, we find that du = -2 sin(2x) dx and v = x. Subsequently, we obtain √(√2/3) [x cos(2x) + 2 ∫ sin(2x) dx]. The integral of sin(2x) dx is straightforward and results in -cos(2x)/2. Hence, the evaluated integral becomes √(√2/3) [x cos(2x) - cos(2x)]. This result incorporates both steps of the integration by parts process: the application of the UV term and the second integral that arises from it.