Integrate cos(4x)+16x^3 with respect to x

This is a simple integration, integrating each individual term with respect to x.For the cos(4x), you should use 'integration by substitution' as it is a function of a function.cos(4x) = cos(u) and u = 4x where dx/du = 4, so dx = (1/4)duso we are now integrating: (1/4)cos(u) duthe 1/4 is a constant so can be taken infront---> integrates to sin(u)The integration of cos(u) is sin(u), using the memorised circle that can be used below:Down is differentiation, up is integration sinxcosx -sinx-cosx (then back to sinx and repeat)so (sin(u))/4and u = 4x so answer is sin(4x)/4Integrating the second value, by adding a power then dividing by the new power:16x^3 becomes (16x^4)/4 = 4x^4So finally, the solution is:sin(4x)/4+ 4x^4+Constant