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a) Calculati limita \( \lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_0^1(x(1-x))^nx^kdx \), unde \( k\in N \).

b) Calculati limita \( \lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_0^1(x(1-x))^nf(x)dx \), unde f e o functie continua pe [0,1].

Asa ca hint, e bine de stiut ca, daca \( I_{n}=\int_0^1x^{n}(1-x)^{n}dx \), atunci are loc formula \( I_{n}=\frac{(n!)^{2}}{(2n+1)!} \).