Question:

<div class='solution'> <p>IBP: $\int\sec^3 x\,dx=\sec x\tan x-\int\sec x\tan^2 x\,dx=\sec x\tan x-\int\sec x(\sec^2 x-1)dx$</p> <p>$=\sec x\tan x-\int\sec^3 x\,dx+\int\sec x\,dx$</p> <p>$2\int\sec^3 x\,dx=\sec x\tan x+\ln|\sec x+\tan x|+C$</p> <p>$\int_0^{\pi/4}\sec^3 x\,dx=\frac{1}{2}[\sec x\tan x+\ln|\sec x+\tan x|]_0^{\pi/4}$</p> <p>$=\frac{1}{2}[(\sqrt{2}\cdot1+\ln(\sqrt{2}+1))-(1\cdot0+\ln 1)]=\frac{\sqrt{2}+\ln(1+\sqrt{2})}{2}$</p> </div>