You can pass the limits to integrate directly:

>>> integrate(1/(x**2+y**2)**Rational(3,2), (y, -L/2, L/2))
L/(x**3*sqrt(L**2/(4*x**2) + 1))

It's generally recommended to do this as it isn't always correct to
substitute the upper and lower values directly.  However, this result
is equivalent to yours in this case, after pulling a 4 out of the
square root.

Aaron Meurer
On Fri, Oct 19, 2018 at 10:17 AM bb <burakbayra...@gmail.com> wrote:
>
> A physics teacher on an online course [presented][1] this integral,
>
> $$
> = \frac{1}{4\pi\epsilon_0} \frac{Q x}{L}
> \int _{-L/2}^{L/2} \left(\frac{dy}{(x^2+y^2)^{3/2}} \right) \hat{x}
> $$
>
> and said she solved it with Wolfram Alpha, which gave
>
> $$
> = \frac{1}{4\pi\epsilon_0} \frac{Q}{x \sqrt{x^2 + (L/2)^2}}\hat{x}
> $$
>
> I was wondering how to solve this using any other symbolic software like 
> Sympy. I tried this for the indefinite integral,
>
>     from sympy import integrate, sqrt, Symbol, pprint
>     y = Symbol('y')
>     x = Symbol('x')
>     print (integrate('1/ ((x**2+y**2)**(3/2))',y))
>
> Result is
>
>     y/(x**3*sqrt(1 + y**2/x**2))
>
> I plugged in the limits,
>
>     from sympy import simplify
>     L = Symbol('L')
>     x = Symbol('x')
>     simplify((L/2)/(x**3*sqrt(1 + (L/2)**2/x**2)) - \
>              (-L/2)/(x**3*sqrt(1 + (-L/2)**2/x**2)))
>
> I get
>
>     2*L/(x**3*sqrt(L**2/x**2 + 4))
>
> which does not look right. Does anyone have any experience solving integrals 
> such as the one above using symbolic software?
>
>   [1]: https://youtu.be/pJwg2Bk0BDE?t=1286
>
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