DJJerome1976

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These are replies submitted by DJJerome1976

@vv Yes, I find it a bit perplexing that Maple attempts the same methods for both, but is only able to evaluate one of them. The rewriting you've done is what is typically taught in a standard integration techniques unit, which facilitates a straightforward substitution. 

Moreover, I observe a similar behavior when attempt to evaluate comparable integrals involving powers of tangent and secant. Risch is being employed, I believe.

trig_integral.mw
 

int(sin(x)^(1/2)*cos(x)^3, x)

-(2/7)*sin(x)^(7/2)+(2/3)*sin(x)^(3/2)

(1)

int(sin(x)^(1/3)*cos(x)^3, x)

int(sin(x)^(1/3)*cos(x)^3, x)

(2)

infolevel[int] := 5

5

(3)

int(sin(x)^(1/2)*cos(x)^3, x)

-(2/7)*sin(x)^(7/2)+(2/3)*sin(x)^(3/2)

(4)

int(sin(x)^(1/3)*cos(x)^3, x)

int(sin(x)^(1/3)*cos(x)^3, x)

(5)

int(tan(2*x)^(1/10)*sec(2*x)^2, x)

(5/11)*tan(2*x)^(11/10)

(6)

int(tan(2*x)^(1/10)*sec(2*x)^4, x)

int(tan(2*x)^(1/10)*sec(2*x)^4, x)

(7)

``


 

Download trig_integral.mw

 

@vv 

I know how to get Maple to perform the integration, the question is why does the second integral require more? By hand, the same approach to integration works, particularly under the assumptions you stated. Other CASs, for example Mathematica, is able to handle both integrals easily.

@Carl Love The standard definition of continuity of a real-valued function, in a first semester calculus course is as follows. A function f:A->B, where A and B are subsets of R, is continous at a in A if:

(1) f(a) exists,

(2) lim f(x) as x->a exists, and

(3) (1) = (2)

Many books refer to this as a `continuity checklist`. This can then be modified using one-sided limits to reflect right- or left-continuity. 

@nm I completely agree with this. How do we make suggestions to the Maple engineers?

@Carl Love Ideally, I would like to restrict the domain and codomain to R. I didn't know is there was a system-wide way of achieving this. This isn't specific to just the sqrt function. It is true also for arcsine and arccosine. I have first-year college students exploring properties such as continuity using Maple. In these classes we use only the real number system, not the complex one. So, some of these results are initially confusing for the students. The follow-up discussion tends to be a good one, however. Thanks for your help!

@Carl Love Is it possible to restrict it to the real domain? 

@acer I like this approach a lot. Originally, I simply wanted the set of possible rational zeros. But the answers have given me more to think about. Much appreciated!

@Kitonum 

Thanks! Just a couple changes, in FullRandom, since you are allowing for a polynomial of arbitrary degree, the if statement should read if nops(q)=N+1. Also, in the definition of S1, it would have to be coeff(q,x,N) so as to grab the coefficient of the leading term.

This is much appreciated!

@tomleslie I am aware of the ability to export graphs from Maple to formats that are easily included in a LaTeX document. That is not what I'm looking for. I want to generate the LaTeX code that produces the desired graphs. Thanks, anyway!

@Kitonum It never dawned on me that shadebetween( ) accepts variable limits. Thanks so much!

@acer yes, this is exactly what I needed. Given how rationalize( ) works, it makes sense that we have to use the reciprocal. Thanks!

@Preben Alsholm This is a nice, efficient solution. I never knew about "freeze." I always seem to learn something here, thanks!

@Kitonum Thanks for the multiple approaches.

@Rouben Rostamian  Thanks for the work around. What's the procedure for reporting a bug? i've never done so.

@Carl Love I am working with matrices with integer entries, so I naturally made the assumption that they are all integers, however, this property does not require integers.

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