If _ n ^ n +1 ~g ( x ) dx = n ^2, n Z , then the value of _ -3 ^3 ~g ( x ) dx is

Step-by-step solution

Given, _n^ n+1 g(x) d x=n^2, n Z at n=-3, _ -3 ⁻² g(x) d x=9 ...(i) at n =-2, _ -2 ⁻¹ g(x) d x=4 ...(ii) at n =-1, _ -1 ^0 g(x) d x=1 ...(iii) at n =0, ₀^1 g(x) d x=0 ...(iv) at n =1 ₁^2 g(x) d x=1 ...(v) at n =2 ₂^3 g(x) d x=4 ...(vi) Adding eqs. (i), (ii), (iii), (iv), (v) & (vi), we get aligned & _ -3 ⁻² g(x) d x+ _ -2 ⁻¹ g(x) d x+ _ -1 ^0 g(x) d x+ ₀^1 g(x) d x+ ₁^2 g(x) d x & + ₂^3 g(x) d x=19 & _ -3 ^3 g(x) d x=19 & aligned