Matematické Fórum

Skrytý text:

is lengthy, but illustrates the connection with the interesting dilogarithmic function. Also, many interesting integrals are computed along the way. I shall omit the obvious computations.

Using the fundamental theorem of calculus and Fubini's theorem, we get
.

We see immediately that .

The is also easy as

and one can use the substitution . This way, you should get .

For , first we use a substitution and compute the inner integral to get
.
Then, substituting , we obtain and after integration by parts, we find
, where .

is the funny one. Using FToC and Fubini once again (on ), we get
, where .

Using Maclaurin's series for and integrating term by term (which is ok for and for we can use the continuity of the integral w.r.t. the upper bound), we obtain .

Thus, it remains to find the value of . Integration by parts gives
,
therefore
.

By putting the results together we find
.