How do you evaluate the integral #int tanxdx# from 0 to #pi# if it converges?
2 Answers
Explanation:
However as
The integral does not converge.
Explanation:
In order to evaluate
and add the results.
# = lim_(brarrpi/2) {: -lncosx]_0^b# (integrate by substitution)
# = lim_(brarrpi/2) (-lncosb + lncos0)#
# = lim_(brarrpi/2) (-lncosb + ln1)#
# = lim_(brarrpi/2) -lncosb#
# = oo# .
The limit does not exist.
So,
Therefore
Bonus material
There is a similar notion, called the Cauchy principal value, that has us, in a sense, evaluate the two improper integrals at once.
For
This limit is
Second example:
The Cauchy principal value is again