I have been trying to prove the following by induction but all my efforts have failed. the original problem is to prove that
$\int_{-1}^{1}(1-x^{2})^kdx = \frac{2^{2k+1}(k!)^{2}}{(2k+1)!}$
Where $k$ is a positive integer.
after expanding the function using the binomial theorem and taking the definite integral I got this
$\int_{-1}^{1}(1-x^{2})^kdx = \sum_{i=0}^{k}\frac {\binom{k}{i}2(-1)^{k-i}}{2 (k-i)+1}$
Now i need to show that $ \frac{2^{2k+1}(k!)^{2}}{(2k+1)!} = \sum_{i=0}^{k}\frac {\binom{k}{i}2(-1)^{k-i}}{2 (k-i)+1}$
I took the base case $k=1$ and it works for $1$
Any help will be much appreciated. Thank you in advance.
I suggest to suppose the thesis valid for $n-1$ and then calculate $$\int_{-1}^1(1-x^2)(1-x^2)^{n-1}dx$$ and integrate by parts using the inductive hypotesis.