_ n [ n (n+1) 2 n+1 + n (n+2) 2(2 n+2) .+ n (n+3) 3(2 n+3) + n terms ]= ₀^1 f(x) d x then f(x)=

Step-by-step solution

Here, aligned & = _ n 1 n _ k=1 ^n n^2 (n+k) k(2 n+k) & = _ n 1 n _ k=1 ^n n^2 (n+k) k(2 n+k) n^2 n^2 & = _ n 1 n _ k=1 ^n 1 ( n+k n ) k n ( 2 n+k n ) aligned aligned & = _ n 1 n _ k=1 ^n 1 (1+ k n ) k n (2+ k n ) & = ₀^1 1 (1+x) x(2-x) d x= ₀^1 f(x) d x aligned f(x)= 1 (1+x) 2 x+x^2