MCQ
$\int_0^{\pi /4} {} \sec x\log (\sec x + \tan x)\,dx = $
- ✓$\frac{1}{2}{[\log (1 + \sqrt 2 )]^2}$
- B${[\log (1 + \sqrt 2 )]^2}$
- C$\frac{1}{2}{[\log (\sqrt 2 - 1)]^3}$
- D$\frac{1}{2}{[\log (\sqrt 2 - 1)]^2}$
✓
Answer
Correct option: A.
$\frac{1}{2}{[\log (1 + \sqrt 2 )]^2}$
a
(a) $I = \int_0^{\pi /4} {\sec x\log (\sec x + \tan x)dx} $
(a) $I = \int_0^{\pi /4} {\sec x\log (\sec x + \tan x)dx} $
Put $\log (\sec x + \tan x) = t \Rightarrow \sec x\,dx = dt$
$ \Rightarrow I = \int_0^{\log (\sqrt 2 + 1)} {t\,dt = \left[ {\frac{{{t^2}}}{2}} \right]} _0^{\log (\sqrt 2 + 1)} $
$= \frac{{{{[\log (\sqrt 2 + 1)]}^2}}}{2}$.
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