Equivalence proof between sum and integral

Given the following sum:

$$\sum_{i=1}^{i=n} \lfloor\log_2(i)\rfloor$$

I want to find a closed form for it, or at least the term of greatest asymptotic growth of its outcome. However, to simplify such procedure, I computed the integral of the summand expression:

$$\int_1^n \log _2(i) \, di=\frac{n (\log (n)-1)}{\log (2)}$$

Then, to verify the correctness of the result, I need to prove that

$$\sum_{i=1}^{i=n} \lfloor\log_2(i)\rfloor=\int_1^n \log _2(i) \, di$$

But, I don't know how to proceed with such task.