please find the mistake

please give the mistake in the following solution:
I=∫ (ax+b)sec x tan x dx4+tan²x dx (a,b>0) and lower limit 0 and upper limit Ï€

splitting the numerator,
∫ax sec x tan x dx3+sec²x = ∫ a(Ï€-x)sec x tan x3+sec²x dx = I₁

so 2I₁= aÏ€ sec x tan x 3+sec²x

So I=(b+ (aÏ€/2)) dt3+t² where t= sec x

integrating we have
I= (b+aÏ€/2)1√3tan^-1 t√3

applying the limits from 1 to -1 as at x=0, t= 1 and at x=Ï€ ,t=-1

we have the answer as -Ï€(aÏ€+2b)6√3

but the answer given is Ï€(aÏ€+2b)3√3

actually in (0,Ï€), the function under the integral is always positive so how can the area under it's curve be negative ......so the negative sign is wrong............but i don't get the mistake in the above solution.......what went wrong ?