Answer
$3{e^{\pi /4}} - 4\sqrt 2 + 1$
Work Step by Step
$$\eqalign{
& \text{let }I=\int_0^{\pi /4} {\left( {3{e^x} - 4\sec x\tan x} \right)} dx \cr
& {\text{Integrate using basic rules }} \cr
& \int {{e^x}} dx = {e^x} + C{\text{ and }}\int {\sec x\tan x} dx = \sec x + C,{\text{ then}} \cr
& I = \left[ {3{e^x} - 4\sec x} \right]_0^{\pi /4} \cr
& {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr
& I = \left[ {3{e^{\pi /4}} - 4\sec \left( {\frac{\pi }{4}} \right)} \right] - \left[ {3{e^0} - 4\sec \left( 0 \right)} \right] \cr
& {\text{Simplify}} \cr
& I= \left[ {3{e^{\pi /4}} - 4\sqrt 2 } \right] - \left[ {3 - 4\left( 1 \right)} \right] \cr
& = 3{e^{\pi /4}} - 4\sqrt 2 + 1 \cr} $$