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I have two vectors, $\bar{x},\bar{v}$ and I want to produce a third vector such that:$$ u_i = \int_{x_i}^{x_i+\delta} f(v_i,t)dt \ .$$ I tried (with a simple function for example):

Integrate[v*(Cos[w*n*t]), {t, x, x +D} ]

but this ends up returning a matrix$$ u_{ij} = \int_{x_i}^{x_i+\delta}f(v_j,t)dt \ .$$ How do I make it so the index of the two vectors is the same? Essentially I want the diagonal of this matrix, but calculating the entire matrix and then taking the diagonal is computationally very slow as the matrix is large. I also tried defining a function

F[i_] = Integrate[Index[v,i]*(Cos[w*n*t]), {t, Index[x,i], Index[x,i] +D} ]

(I could not access vector elements in the function definition using x[[i]]) but this does not evaluate when I enter say F[1].

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Here is a simple example with a particular x and v and f:

v = {1, 2, 3}; x = {4, 5, 6}; 
f[v_, t_] := Cos[v + t]; 
Table[Integrate[f[v[[i]], t], {t, x[[i]], x[[i]] + d}], {i, Length[x]}]

This returns three components. Alternately you can bypass the use of Table:

Integrate[f[v[[#]], t], {t, x[[#]], x[[#]] + d}] & /@ Range[Length[x]]
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