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# Tag Archives: Integral

## Exponential fractional part

Let denote the fractional part of . Evaluate the integral

**Solution**

By definition it holds that

hence

We note however that

Hence,

Summing up,

## Arctan squared integral

Evaluate the integral

**Solution**

First of all we note that for

Hence by Parseval we get that

## Contour radical integral

Consider the branch of which is defined outside the segment and which coincides with the positive square root for . Let then evaluate the contour integral:

**Solution**

It is a classic case of residue at infinity. Subbing the counterclockwise contour integral rotates the northern pole of the Riemannian sphere to the southern one and the contour integral is transformed to a clockwise one. Hence:

The equality **does hold** for all if we take the standard branch , otherwise it is not that obvious why this holds, since we are dealing with a multi-valued function.

## Square logarithmic integral

Prove that

**Solution**

**Lemma 1: **Let . It holds that

**Lemma 2: **Let . It holds that

We begin by squaring the identity of lemma 2. Hence,

Integrating the last equation we get,

Expanding the LHS we get that

Finally,

## A parametric logarithmic integral

Let . Prove that

**Solution**

We are basing the whole solution on the Beta function and its derivative. We recall that

(1)

Setting and back at we get that

Differentiating with respect to we get that

where we made use of the reflection formulae of both the Gamma and the digamma function; and .

Now for our integral we have successively: