^2 x ( x) ^ x dx = - Vidyadip

MCQ

$\int {\frac{{{{\sec }^2}x}}{{\log {{(\tan x)}^{\tan x}}}}dx = } $

A
$\log \left| {\log {{(\tan x)}^{\tan x}}} \right| + c$
B
$\log (\tan x) + c$
✓
$\log \left| {\log (\tan x)} \right| + c$
D
$\log \left| {\frac{{\log \tan x}}{{\tan x}}} \right| + c$

✓

Answer

Correct option: C.

$\log \left| {\log (\tan x)} \right| + c$

c
Put tan $x=t$

$\sec ^{2} x d x=d t$

$\int {\frac{{{\rm{dt}}}}{{\ln {{({\rm{t}})}^{\rm{t}}}}}} = \int {\frac{{{\rm{dt}}}}{{{\rm{t}}\ln {\rm{t}}}}} $

Put $\ln t = z$

$\frac{{{\rm{dt}}}}{{\rm{t}}} = {\rm{dz}}$

$=\int \frac{\mathrm{d} z}{z}$

$=\ln z+c$

$ = \ln \ln {\rm{t}} + {\rm{c}}$

$ = \ln \ln \tan x + c$

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