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

MCQ

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

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)
Put $1+\tan x= t \Rightarrow \sec ^2 x d x= dt$
When $x=0, t =1$ and when $x=\frac{\pi}{4}, t =2$
$\therefore \quad \int_0^{\pi / 4} \frac{\sec ^2 x}{(1+\tan x)(2+\tan x)} d x$
$\begin{array}{l}=\int_1^2 \frac{d t}{t(1+t)}=\int_1^2 \frac{d t}{t}-\int_1^2 \frac{d t}{1+t} \\ =[\log t-\log (1+t)]_1^2\end{array}$
$\begin{array}{l}=\log _e 2-\log _e 3+\log _e 2 \\ =\log _e\left(\frac{4}{3}\right)\end{array}$

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