To begin, let's first apply the sum rule to break #intcosx-sec^2xdx# into #intcosxdx-intsec^2xdx#. Now we can evaluate these integrals one by one:
#intcosxdx=sinx+C-># (the derivative of sine is cosine, so the antiderivative of cosine is sine)
#intsec^2xdx=tanx+C-># (same logic)
The whole solution is: #f(x)=sinx+C-tanx+C#. Because #C# is just some constant, #C+C# is just another constant, so we can write: #f(x)=sinx-tanx+C#. To solve for #C#, we use the given initial condition that #f(pi/8)=-1#:
#-1=sin(pi/8)-tan(pi/8)+C#
#-1=(sqrt(2-sqrt(2)))/2-(sqrt(2)-1)+C#
#0=(sqrt(2-sqrt(2)))/2-sqrt(2)+2+C#
#C=-(sqrt(2-sqrt(2))-2sqrt(2)+4)/2~~-0.968#
Thus, #f(x)=sinx-tanx-(sqrt(2-sqrt(2))-2sqrt(2)+4)/2#, or, using an approximation, #f(x)=sinx-tanx-0.968#.
