Hi Catherine!
On Thu, Mar 18, 2010 at 1:23 PM, Catherine Kehl <[email protected]> wrote:
> 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.)
This is what I got:
In [2]: a = 5; n = 7; c = 11; d = 13
In [3]: integrate((a**2-x**2)**n, (x, c, d))
Out[3]:
5757254575990452224
- ───────────────────
6435
In [4]: (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)*integrate((a**2-x**2)**(n-1), (x, c, d))
Out[4]: -894678255787172.
So in [4] I got a different number than you did. SymPy should be able
to do most of the stuff from calculus books, so I'll investigate this
later, unless someone beats me to it.
Ondrej
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