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# Tag Archives: Integral

## Trigonometric integral

Prove that

Solution

Let be the integral. Note that

and hence:

For the integral we apply the substitution . Then, and

(1)

and similarly by applying the change of variables at the second integral we get that

(2)

Adding equations , we get that

and the result follows.

## Parametric integral

Let . Prove that:

Solution

We’re applying the change of variables and thus,

## Abel’s Integral

Prove that:

Solution

We are applying Abel – Plana. We choose thus,

## Ahmed’s Integral

Prove that

Solution

Consider the function . Differentiating with respect to we have that:

We integrate the last equation from to . Thus,

However,

Hence the last equation gives

(1)

Suffice to calculate the integral. Applying the change of variables we have:

Going back at we have that:

## Coxeter’s Integral

Prove that

Solution

We state lemmata:

Lemma 1: It holds that .

Lemma 2: It holds that .

Proof: We have successively:

Lemma 3: It holds that where .

Proof: We have successively:

We are ready to attack the initial monster. For that we have:

### Who is Tolaso?

Find out more at his Encyclopedia Page.