First slide

Properties of ITF

Question

If $\log_{2} x \geq 0$ , then $\log_{1 / π} \{\sin^{- 1} \frac{2 x}{1 + x^{2}} + 2 \tan^{- 1} x\}$ is equals

Moderate

A

$\log_{1 / π} (4 \tan^{- 1} x)$
B

0
C

$- 1$
D

none of these

Solution

We have,
$\log_{2} x \geq 0 \Rightarrow x \geq 2^{\circ} = 1$ .
For $x \geq 1$ .we have
$\begin{array}{l} \sin^{- 1} (\frac{2 x}{1 + x^{2}}) = π - 2 \tan^{- 1} x . \\ ∴ \log_{1 / π} \{\sin^{- 1} \frac{2 x}{1 + x^{2}} + 2 \tan^{- 1} x\} \\ = \log_{1 / π} \{π - 2 \tan^{- 1} x + 2 \tan^{- 1} x\} \\ = \log_{1 / π} π = - 1 . \end{array}$

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