find f. f=_______

Question

f '(t) = 8 cos(t) + sec²(t),    −π/2 < t < π/2,    f(π/3) = 3

Answer 1

If f '(t) = 8 cos(t) + sec²(t) then f(t) = ∫8 cos(t) + sec²(t) dt

= ∫8 cos(t) + sec²(t) dt

= ∫8 cos(t) dt + ∫sec²(t) dt

= 8sin(t) + tan(t) + C

Since f(π/3) = 3 then

3 = 8sin(π/3) + tan(π/3) + C

3 = 8(√3/2) + √3 + C

3 = 4√3 + √3 + C

C = 3 - 5√3

f(t) = 8sin(t) + tan(t) + 3 - 5√3