## A definite integral

Let be a continuous function such that

(1)

Evaluate the integral .

Solution

First of all we note that the equation

(2)

has a unique root , lets call it . Hence and thus . We note that is rewritten as

(3)

Integrating from to we get

## Periodicity and integral

Let be a continuous and periodic function with period . If then prove that:

1. where .

Solution

1. We have successively:

2. We have successively:

## On a geometric sequence

Let be four consecutive terms of a geometric sequence. Prove that

Solution

We have successively:

## Constant area

Let be a positive real number. The parabolas defined by and intersect at the points and .

Prove that the area enclosed by the two curves is constant. Explain why.

Solution

First of all we note that

Hence,

## Binomial sum

Let . Evaluate the sum

Solution

We have successively