Matematické Fórum

Skrytý text:
We express the function, which is supposed to be integrated, as a linear combination of odd and even functions. Changing the limits of integration of such functions causes that a lot of integrals vanish.

Firstly, remind that



Define two functions:



It is not hard to show that is an odd function and is an even function, thus



and moreover,



and



Hence



Now we show that . Keeping in mind the fact that is an odd function, determine the integral :



Similarly determine the integral :



Finally, we determine the integral :



We can see that the original integral equals to , i.e.