Using conservation of energy (let maximum of be ) you must have:

but since we have

so

which is of separable variable form so

.

Let then so this becomes

.

At this point we have answered the question as the integral has no dependence on so is simply a numerical constant and so we've shown that the time period is inversely proportional to the amplitude .

If you must know what the integral evaluates to:

,

so not anything nice in terms of elementary functions!