Help calculating two integrals-- generalized and definite 
The first one is to calculate
$$\int_{-\infty}^{\infty} \frac{1}{(1+4x^2)^2} dx.$$

I think this one should be solvable with the method of substitution but I tried using $t=4x^2$ which didn't work well, or where I have miscalculated something.

The second one is, for every $n\in\mathbb N$, to calculate
$$\int_{-1}^{1} (1-x^2)^n dx.$$

So, I really don't know how to solve the second integral. Maybe through the use of $n=1,2,3,\cdots$ and look for any patterns?

The first integral can also be evaluated using Glaisher's theorem, which says that if:
$$f(x)=\sum_{n=0}^{\infty}(-1)^n c_n x^{2n}$$
then we have:
$$\int_{0}^{\infty}f(x) dx = \frac{\pi}{2}c_{-\frac{1}{2}}$$
if the integral converges and where an appropriate analytic continuation of the series expansion coefficients has to be used (e.g. factorials replaced by gamma functions). This is a special case of Ramanujan's master theorem.
In this case we can easily obtain the series expansion. Differentiating the the geometric series:
$$\frac{1}{1+u} = \sum_{n=0}^{\infty}(-1)^n u^n$$
yields:
$$\frac{1}{(1+u)^2} = \sum_{n=0}^{\infty}(-1)^n (n+1)u^n$$
We thus have:
$$\frac{1}{(1+4 x^2)^2} = \sum_{n=0}^{\infty}(-1)^n (n+1)4^n x^{2n}$$
This means that $c_n = (n+1)4^n$ and $c_{-\frac{1}{2}} = \frac{1}{4}$, the integral from minus to plus infinity is thus $\frac{\pi}{4}$.
