Question
$\int_0^{\pi /4} {\frac{{{{\sec }^2}x}}{{(1 + \tan x)(2 + \tan x)}}} \,dx = $
$\therefore \,\,\,\int_0^{\pi /4} {\frac{{{{\sec }^2}x}}{{(1 + \tan x)(2 + \tan x)}}dx} $
$ = \int_1^2 {\frac{{dt}}{{t(1 + t)}}} = \int_1^2 {\frac{{dt}}{t} - \int_1^2 {\frac{{dt}}{{1 + t}}} } = [\log t - \log (1 + t)]_1^2$
$ = {\log _e}2 - {\log _e}3 + {\log _e}2 = {\log _e}\frac{4}{3}$.
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