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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,