Evaluate: x 2 x 3 x d x - Vidyadip

MCQ

Evaluate: $\int \tan x \tan 2 x \tan 3 x d x$

A
$\frac{1}{3} \log |\sec 3 x|-\log |\sec x|+C$
B
$\log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|+C$
C
$\log |\sec x|-\frac{1}{2} \log |\sec 3 x|+\frac{1}{2} \log |\sec 2 x|+C$
✓
$\frac{1}{3} \log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|-\log |\sec x|+C$

✓

Answer

Correct option: D.

$\frac{1}{3} \log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|-\log |\sec x|+C$

Let $I=\int \tan x \tan 2 x \tan 3 x d x$
$\text { Since, } \tan 3 x=\tan (2 x+x)=\frac{\tan 2 x+\tan x}{1-\tan x \tan 2 x}$
$\Rightarrow \tan x \tan 2 x \tan 3 x=\tan 3 x-\tan 2 x-\tan x ...(i)$
$\therefore I=\int(\tan 3 x-\tan 2 x-\tan x) d x \ ($From $(i))$
$=\frac{1}{3} \log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|-\log |\sec x|+C$

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