Question

Evaluate the integral.[0,pi/4]∫( sec(t)tan(t)+ tcos(2t)j+ ((sin(2t))^2) *(cos(2t))k) dt

Answer 1

do the j and k integrals separately
∫sect tant dt = sect
∫t cos(2t) dt: use integration by parts, with u=t, dv = cos(2t) dt
∫sin^2(2t) dt = ∫(1-cos(4t))/2 dt

see what you can do with that. You can always check your work at wolframalpha.com.