Can we express the integral of the nth derivative of this function analytically?

Question

I am currently working on an assignment with Legendre Polynomials. The integral I get stuck with is in fact the integral of the Legendre Polynomial itself i.e. $$\int \frac{1}{2^n n!} \frac{d^n}{dt^n} (t^2 - 1)^n dt$$ I am tempted to integrate this naively but I feel as though that would not be the solution. The initial function that I am to approximate using the Legendre polynomials is a piece-wise function where $f(t) = 1$ for $t \in [0, 1]$ and $f(t) = -1$ for $t \in [-1,0]$.

Answer 1

The answer is of course: $$ \int \frac{1}{2^n n!}\frac{d^n}{dt^n}(t^2-1)^n dt = \frac{1}{2^n n!} \frac{d^{n-1}}{dt^{n-1}}(t^2-1)^n$$ What to do after that remains a mystery...