First slide

Properties of ITF

Question

If $2 \tan^{- 1} x + \sin^{- 1} (\frac{2 x}{1 + x^{2}})$ is independent of $x,$ then

Moderate

A

$x \in [1, \infty) \in (- \infty, - 1)$
B

$x \in [- 1, 1]$
C

$x \in (- \infty, 1]$
D

none of these

Solution

We know that
$\sin^{- 1} (\frac{2 x}{1 + x^{2}}) = \{\begin{cases} 2 \tan^{- 1} x, if - 1 \leq x \leq 1 \\ π - 2 \tan^{- 1} x, if x \geq 1 \\ - π - 2 \tan^{- 1} x, if x \leq - 1 \end{cases}$
$∴ 2 \tan^{- 1} x + \sin^{- 1} \frac{2 x}{1 + x^{2}} = \{\begin{cases} 4 \tan^{- 1} x, if - 1 \leq x \leq 1 \\ π, if x \geq 1 \\ - π, if x \leq - 1 \end{cases}$
Thus, $2 \tan^{- 1} x + \sin^{- 1} \frac{2 x}{1 + x^{2}}$ is independent of $x,$ if
$x \in [1, \infty)$ or, $x \in (- \infty, - 1]$ .

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