If d x 4 x^2-1 =A (2 x-1 2 x+1 )+c, then A= ........ - Vidyadip

MCQ

If $\int \frac{d x}{4 x^2-1}=A \log \left(\frac{2 x-1}{2 x+1}\right)+c,$ then $A= ........ $

A
$1$
B
$\frac{1}{2}$
C
$\frac{1}{3}$
✓
$\frac{1}{4}$

✓

Answer

Correct option: D.

$\frac{1}{4}$

$\frac{1}{4}$
$\int \frac{d x}{4 x^2-1}=A \log \left(\frac{2 x-1}{2 x+1}\right)+c ...\ ($given$)$
$\therefore \int \frac{d x}{(2 x)^2-1}= A \log \left(\frac{2 x-1}{2 x+1}\right)+c$
$\therefore \frac{1}{2(1)(2)} \log \left(\frac{2 x-1}{2 x+1}\right)+c$
$= A \log \left(\frac{2 x-1}{2 x+1}\right)+c$
$\therefore A =\frac{1}{4}$

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