I'll assume the question is #f(1)# where #f(x) = x arctan x#.
#f(1)= (1) (arctan(1)) = arctan 1 #
Normally I'd treat the #arctan# as multivalued. But here with the explicit function notation #f(x)# I'll say we want the principal value of the inverse tangent. The angle with tangent 1 in the first quadrant is #45^circ# or #pi/4#:
#f(1)= (1) (arctan(1)) = arctan 1 = pi/4#
That's the end. But let's put the question aside, and focus on what #arctan t# really means.
I usually think of #tan ^-1(t)# or equivalently (and I think better notation) #arctan(t)# as a multivalued expression . The "function" arctan isn't really a function, because it is the inverse of something periodic, which can't really have an inverse over its entire domain.
This is really confusing for students and teachers. All of a sudden we have things that look like functions that aren't really functions. They've kinda slipped in under the radar. New rules are required for dealing with them, but they're never explicitly stated. Math starts to get fuzzy when it shouldn't.
#x=arctan t# is best thought of as the solutions to #tan x=t.# There are a countably infinite number of them, one per period. Tangent has period of #pi# so the solutions are #pi# apart, which is where the #pi k# comes from, integer #k#.
I usually write the principal value of the inverse tangent as Arctan, with a capital A. Unfortunately Socratic keeps "correcting" it. I'll fudge it here:
#t = tan x# has solutions
#x = arctan t = text{Arc}text{tan}(t) + pi k quad# for integer #k#.
