_0^ /4 ^2 x (1 + x)(2 + x) ,dx = - Vidyadip

MCQ

$\int_0^{\pi /4} {\frac{{{{\sec }^2}x}}{{(1 + \tan x)(2 + \tan x)}}} \,dx = $

A
${\log _e}\left( {\frac{2}{3}} \right)$
B
${\log _e}3$
C
$\frac{1}{2}{\log _e}\left( {\frac{4}{3}} \right)$
✓
${\log _e}\left( {\frac{4}{3}} \right)$

✓

Answer

Correct option: D.

${\log _e}\left( {\frac{4}{3}} \right)$

d
(d) Put $1 + \tan x = t \Rightarrow {\sec ^2}x\,dx = dt$

$\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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