Matematické Fórum

Skrytý text:

The given integral can be transformed and is equal to . It is known that the function can be expressed as certain sum of cosines (Dirichlet kernels in particular) and therefore it's relatively easy to integrate.

Skrytý text:

Let , substitute and use sine's properties to get
(now, one may notice that for but that's not our case). Make another substitution to get . Now, I shall illustrate one way how to reduce the function into the sum of Dirichlet kernels . At first observe that holds. Then sum this equality with alternating sign and k decreasing to cancel most of the terms and obtain . We need to integrate this equation, so let . Now there are number of ways how to compute the preceding sum, one can make groups of two or four to cancel the terms in the kernels. For instance, observe that (*) , thus and in case of we have . It follows from (*) that , hence the result. (The sum is partial sum of well known Leibniz series for ).