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You've taken the wrong limits, the limits are 5π/16 and π/16.Alright... Here you go...:
Integration of
(1 + sec2 4x) dx----> x+ tan 4x/4
For limit: π/16
π/16 + 1/4 tan π/4 = π/16 +1/4
For limit: -π/16
-π/16 +1/4 tan -π/4 = -π/16 - 1/4
Subtracting both: [ π/16 +1/4] - [ -π/16 - 1/4]
= 2 π/16 + 1/2
π/8 +1/2 = 1/8 ( π + 4) ----- k= 1/8