For a National Board Exam Review:
Evaluate:
$${ \int \cot^3(ax) csc^2(ax)dx }$$
Answer is
$${ -\frac{1}{4a}cot^4ax +C }$$
Ok. I certainly know how to integrate it from scratch. What I'm doing is like the calculator/brute force method: You put some arbitrary limit say lower_limit = 0.5, upper_limit = 2.5, replace constants like here "a" with 1 or something. Then among the choices from the set evaluate it according to difference between the given limits.
This may seem unmathematical... and yes, it does not always work like in this case... but why do I prefer the calculator method?
National Board Exam for Mechanical Engineering is just too broad for us to memorize everything from Algebra to Power Plants. Review center told us that if given a really complex integral; don't try to integrate.
If given a complex integral, yes, it might actually take longer. But it gives you the right answer; and no memorizing. Best to play safe if you are going to spend precious time on it anyway. So just handle these types when you've done all the harder problems. Use it only when needed.
Ok so I do try to evaluate this using the limits and values given above... and I do get a math error. I think it has something to do with the cosecant function... If i try anything higher than 1... or like pi/4.
What should I do to get around this?
$\endgroup$ 61 Answer
$\begingroup$You tell us neither what your "math error" is nor how you did the calculations. But here is a screenshot of following your approach on the TI-Nspire CX graphing calculator, commonly used in high schools (such as where I teach) and in some colleges.
As you can see, there is no error here. There would have been errors if I mis-typed the integral. One common error is forgetting the multiplication sign in the subexpression $a\cdot x$: without that sign, the calculator thinks you are trying to use the two-letter variable $ax$. Another common error is trying to exactly copy the math notation $\cot^3 a\cdot x$. The calculator requires parentheses around the argument $a\cdot x$, and it does not like the cube symbol just after the function name. It must be typed as $\cot(a\cdot x)^3$ (which is how I typed it) or as $(\cot(a\cdot x))^3$ (which is what the calculator converts it to).
The numeric check usually works, if you keep set the integral limits so the variable stays in the domain of the integrand. In your case the domain includes $(0,\pi)$, so your limits of $0.5$ and $2.5$ are just fine. A graphing calculator can be confused in some cases, so you should not completely trust it, but numeric checks like this are very useful. I emphasize such numeric checks in my calculus class. In fact, our textbook is titled Calculus: Graphical, Numerical, Algebraic, and I do teach multiple approaches to solving problems, using them to check each other. I also give exercises that show that the calculator is not to be completely trusted: it can be fooled. Syntax and other details must also be carefully observed, of course. So the calculator is not perfect, but it is a useful tool.
As for your overall point, using numeric methods rather than algebraic/analytic ones is (usually) valid, especially for multiple choice questions. Wisdom must be used as to when this is possible and when it may actually save time.
I should state another way to check each multiple choice answer. Since you are looking for an indefinite integral, compare the graph of the derivative of each choice with the graph of the integrand. You may need to zoom in the graph to see the graphs more clearly. This is what I get for your problem and possible answer. You can't see it here, but the red graph exactly lies over the blue graph, showing the graphs are the same.
