## Integral inequality (I)

Let be a differentiable and convex function such that and . Prove that .

## Symmetry

Let be continuous functions. If is symmetric around the line then prove that:

## An integral from a geometric view

Evaluate the integral

Solution

## 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: