i) differentiate xcos2x with respect to x ii) integrate xcos2x with respect to x

i) For this question we need to use the product rule as there are two functions that are being multiplied together Product Rule: (uv)'=u'v + v'uApply this formula to the problem: xcos2x => u=x, u'=1 => v=cos2x, v'= -2sin2x -> Differentiation of trigonometric functions u'v + v'u = 1 x cos2x + - 2sin2x x x = cos2x - 2xsin2xii) For this question we must use integration by parts as there are two functions that have been multiplied together Integration by parts: ∫udv = uv - ∫vdu Apply this formula to the problem: ∫xcos2x => u= x, du=1 => v=1/2sin2x, dv=cos2x *v was found by integrating dv Insert these into the formula:∫udv = x x 1/2sin2x - ∫1/2sin2x x 1 = 1/2xsin2x - ∫1/2sin2x =1/2xsin2x - - 1/4cos2x + c -> Always remember to add a constant when integrating with no limits =1/2xsin2x + 1/4cos2x + c