This is a specific problem, rather than an interesting theory
question, but I'm hoping someone can help untangle me a bit.
A hobby of mine the last bit has been working through Apostol's
Calculus, and thereby remedying some of the deficits in my math
education. (This is a very inexpensive hobby, amortizing the cost of
the books over the time you put into it, if you do all the problems.)
Anyhow, while banging my head against, this problem*:
Use integration by part to derive the following formula:
\int(a^2-x^2)^n dx = \frac {x(a^2-x^2)} {2n+1} + \frac {2a^2n} {2n+1}
\int(a^2-x^2)^{n-1} dx + C
(I'm hoping tex is enough of a lingua franca to pass muster here? This
is 5.10.15a, on page 221...)
Anyhow, having not found my way through this, I decided to run it
though sympy as there are a (very) few misprints in Apostol... and
hey, I've been wanting to play more with sympy.
Having set a=5, and n=7, c=11 and d=13 I gave it:
In [29]: s.integrate((a**2-x**2)**n, (x, c, d))
to which it returned:
Out[29]: -5757254575990452224/6435
I then fed it:
In [31]: (d*(a**2-d**2)**n)/(2*n+1)-(c*(a**2-c**2)**n)/(2*n
+1)+(2*a**2*n)/(2*n+1)*s.integrate((a**2-x**2)**(n-1), (x, c, d))
for which I received:
Out[31]: -2693709139701405314/3003
(You will note that these values, while vaguely close, are not equal.)
Following the time honored tactic of asking the geek to one's left, I
ran it past my roommate, who is generally more of a python head than
I. He ran it through Mathematica with the same values, and results
that equaled my first result, and which equaled each other. And yet
did not find any issues with my code, and did not produce different
results from mine with minor variants on my code. (I've also tried
other values, but only these were tested in another system.)
So yeah, I'm home sick, but I'm no longer feverish and I still can't
see why these should not be equivalent. (Not that I'm overly fond of
Wolfram, but when Wolfram and Apostol agree...)
Help?
Catherine
* I am not asking for help on the problem, mind - I would really
prefer to keep banging my head against it until it or my head gives
way.
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