## Evaluation of integral [MAA]

Let be a continuous real valued function on satisfying the identity

Evaluate the integral

*(D.M.Batinetu-Giurgiu , George Emil Palade)*

**Solution**

## On an exercise with classic questions

Let be a twice differentiable function such that forall . If has a local extrema at and the value there is and it holds that

then

(i) Evaluate the area bounded by the graph of , the lines and the axis .

(ii) Let . If

holds then prove that

(iii) Prove that is invertible in . If you also know that the graph passes through the points and then evaluate the value of

(iv) Prove that forall it holds that

**Solution**

Since forall we conclude that is convex. We also conclude that is strictly increasing in . Since has a local extrema at this has to be a local minimum, because (it follows from Fermat’s theorem) and combining the monotony along with the root of the derivative we get that

(i) The area bounbed by the graph of , the lines and the axis is equal to

since .

(ii) Successively we have

(iii) We begin by the classical change of variables thus:

(iv) We simply note that

and the result follows.

## An inequality using AM – GM

Let be positive numbers. Prove that

**Solution**

and we conclude the result.

## An integral with Gamma and digamma

Let denote the Euler’s Gamma function and denote the digamma function. Evaluate

**Solution**

The exercise can also be found at Aops.com .

## An inequality involving harmonic number

Let denote the – th harmonic number and let . Prove that

**Solution**

We might begin with the integral representation of the harmonic number, namely the equation:

So we have to prove the equivelant inequality

and this is obvious using the Cauchy – Schwarz inequality.

The exercise can also be found here .