Answer
$$f\left( s \right) = 4\sec s + 1 - 4\sqrt 2 $$
Work Step by Step
$$\eqalign{
& f'\left( s \right) = 4\sec s\tan s;{\text{ }}f\left( {\pi /4} \right) = 1 \cr
& {\text{Calculating the general solution}} \cr
& f\left( s \right) = \int {f'\left( s \right)} ds \cr
& f\left( s \right) = \int {\left( {4\sec s\tan s} \right)} ds \cr
& f\left( s \right) = 4\sec s + C \cr
& {\text{Calculating the particular solution for }}f\left( {\frac{\pi }{4}} \right) = 1 \cr
& 1 = 4\sec \left( {\frac{\pi }{4}} \right) + C \cr
& 1 = 4\sec \left( {\frac{\pi }{4}} \right) + C \cr
& 1 = 4\sqrt 2 + C \cr
& C = 1 - 4\sqrt 2 \cr
& {\text{The particular solution is}} \cr
& f\left( s \right) = 4\sec s + 1 - 4\sqrt 2 \cr
& {\text{Graphing general solutions for }}C = - 1,{\text{ 2, 3 and the particular}} \cr
& f\left( s \right) = 4\sec s + 1 - 4\sqrt 2 \cr} $$
