Below is an attempt to evaluate an integral from t to ZERO.  I believe the issue is that sin(x) in the denominator is causing a problem.  However, in actuality sin(nx) is in the numerator so at the lower integral bound approaching ZERO the factor of n remains.  So how can I get MAPLE to evaluate this correctly?
 

Download evaluating_integral.mw

In the link below I attempt to solve 2 trig series which are essentially equivalent as indicated by the numerical output of eq (5).  The series  represented by S13 & S14 has arguments of the trig functions that realizes that only the odd terms for k yield non-zero results.  The case represented S11 & S12 by makes no such presumption; nonetheless, all cases agree within reason numerically.  Now to find min/max values taking the derivative is needed which is simply done by removing the integral as indicated by Q1 through Q6.

Now resolving the roots works OK for Q6 because beta = 2*pi *t/T conveniently collapsed the numerator into factorable expressions.  Resolving the roots for Q3 did not work so well because what I think is that the expression in red has multiple roots so it only spits out t as the solution?  I expressed the angle alpha in terms of beta & probably need to resolve kappa to somehow get the expression in red to collapse into a factored expression, but I am not sure how to execute this.  When I solve for kappa I get ZERO.

Does anyone have suggestions?  Remember I demonstrated that both series are practically idendical numerically; hence, there derivatives should be as well as long as both series are well behaved functions.  So the solutions must be the same as well.

trig_series_solns.mw

I have a series with an integral inside the series.  I have worked the problem 2 different ways using sum vs Sum.  The integration variables are independent of the series variables so swapping the order of operation should not matter, but in the case, (S2), I do get a difference & I do not understand why.  The explanation fo INERT vs ACTIVE I do not think explains this.  The reason why I say this is because S1 the ACTIVE sum concurs with the INERT expressions S3 & S4.  S2 is swapping the order of operation for the ACTIVE sum does not yield the same result as the other 3 cases.  Why is this?  I am at a loss so including examples would be helpful to me.

swapping_orders_of_operation.mw

See below, why the difference + what is the distinction between the 2?  Notice that they yield different results with the is command.

Download sum_versus_Sum.mw

Below is the output from test relation that seems contradictory.  All I did was swap the order of operation from summing an integral to taking the integral of the sum.  The summation and integration variables are independent so I would think the statement for S4 would yield a TRUE result.  Also, I got a FALSE return on a well known trig identity.

Am I missing something subtle or even obvious?

equivalence_discrepancy.mw