the integrand does not have an anti-derivative that can be expressed in terms of standard functions (visit convince yourself that our claim is right note that wolfram alpha does give you an answerbut that answer is an approximationit is not exact this is because wolfram alpha too uses numerical methods to arrive at the answerjust as you will in this exercisethereforewe are forced to compute the integral by numerical methods compute result that is right to four digits hint use ideas from exercise filenameintegrate_x py exercise integrate products of sine functions in this exercise we shall integrate ij; sin jxsin kx/dx where and are integers aplot sin xsin xand in separate plots exr sin xsin xfor plain why you expect sin sin dx and sin sin dx buse the trapezoidal rule to compute ij; for and filenameproducts_sines py exercise revisit fit of sines to function this is continuation of exercise the task is to approximate given function ton oeby sum of sinessn td bn sin nt( nd we are now interested in computing the unknown coefficients bn such that sn tis in some sense the best approximation to tone common way of doing this is to first set up general expression for the approximation errormeasured by "summing upthe squared deviation of sn from ed sn // dt we may view as function of bn minimizing with respect to bn will give us best approximationin the sense that we adjust bn such that sn deviates as little as possible from