Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer.
$a = 2$
$a = 4$
$a = 6$
Now, a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an odd, positive integer.
$a = 1$ (Common)
$a = 3$
$a = 5$
$a = 7$
The integrals themselves are hideous.. But the graphs are fun to observe (At least for me), especially for the cases where a is odd.
I know that the difference in these graphs (Between odd and even $a$ ) is caused mainly by the powers the sin and cos functions are raised to, rather than the powers of the arguments of these functions.
I have 4 questions :-
Why does $\int \sin^a(x^a)\cos^a(x^a)dx$ , where $a$ is even, have fewer and fewer oscillations (i.e. the line becomes less wavy) as the value of $a$ increases? (Note: I know that the lines come closer and closer to the x-axis as evidenced by the outputs of the integral on the graph)
What is going on when $a$ is odd? Can you explain why that beaker-like structure is formed; specifically, why does it seems to dip around a certain value ,then raise again to oscillate so much and the die out?
Some questions regarding an observation of the case $a = 5$: Why does there seem to be a tiny bump just after the '4' on the x-axis? What value could be causing this, and why?
And when $a = 7$, why do there seem to be tiny successive bumps beyond 2 and -2?
I realize that answers may not exist for all of these questions, because these maybe the intrinsic qualities of the graph of the integral in discussion, but I'm hoping that some people may have insights and explanations for the features of the graphs I've questioned about.... Especially questions 3 and 4. I hope you don't consider the question silly; I'm genuinely interested in knowing the reasons (If they exist) for the above graphs.