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# Tag Archives: Integral

## Logarithmic integral

Evaluate the integral

## An arcosine integral

Evaluate the integral

## A limit

For any nonnegative integer , define

Evaluate the limit .

**Solution**

**Lemma: **Let and . It holds that

*Proof: *The LHS is just the imaginary part of

This is a geometric progression , hence:

The result follows.

Hence,

Now,

## Fractional integral

Let denote the fractional part. Prove that

for the different values of the integer number .

**Solution**

Let denote the integral,

since if whereas if . Therefore,

except of a countable set whose measure is .

## Multiple logarithmic integral

Let denote the Riemann zeta function. Evaluate the integral

**Solution**

Based on symmetries,

Let . It follows that

Using the recursion we get that

Thus,