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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,

## 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: