intergrate xcos(2x) with respect to x

The function of x has two separate x terms multiplied by each other. Therefore, the appropriate way to integrate this function is by parts. The equation for integration by parts is [integral (u.dv/dx)] = uv - [intergral (v.du/dx)]. The first step of integration by parts is to decide which term is the "u" term and which term is the "dv/dx" term. The best way to decide this term is to see which function you would rather differentiate to make the integral easier. From this question it can be seen that the "x" term would be easier as it differentiates to 1. A general rule of thumb is to use the acronym LATE to decide "u". This describes the order of preference for deciding your "u" function. L - logarithmic, A - algebraic, T - trigonometric, E- exponential. As A come before T it is clear to see that "x" should be the u and cos(2x) should be the dv/dx term. Once that is decided then you find the du/dx and v terms and plug into the equation to get: (xsin2x)/2 - [integral sin2x/2]. You then integrate the second function to get a final answer of: (xsin2x)/2 + cos2x/4 (+ C). This can be simplified to: 1/4(2xsin2x + cos2x + D) where D = C/4.