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Intersection of Sets

46. If log_x+...

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46. If }\operatorname{log}_{x+1}2x-1+\operatorname{log}_{2x-1}x+1=2, find }x

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A (x) = ∣ ∣ ∣ ∣ \begin{matrix} x + 1 & 2 x + 1 & 3 x + 1 2 x + 1 & 3 x + 1 & x + 1 3 x + 1 & x + 1 & 2 x + 1 \end{matrix} ∣ ∣ ∣ ∣

\int_{0}^{1} A (x) d x =

x \geq 1, t h e n 2 T a n^{- 1} x + S i n^{- 1} (\frac{2 x}{1 + x^{2}})

x \geq 1 t h e n 2 t a n^{- 1} x + s i n^{- 1} (\frac{2 x}{1 + x^{2}})

=

x \geq 1, t h e n 2 T a n^{- 1} x + S i n^{- 1} (\frac{2 x}{1 + x^{2}})

Q. Prove the following

(1) {sin}^{- 1} (\frac{2 x}{1 + x^{2}}) = 2 {tan}^{- 1} x, | x | \leq 1

(2) {cos}^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) = 2 {tan}^{- 1} x, x \geq 0

(3) {tan}^{- 1} (\frac{2 x}{1 - x^{2}}) = 2 {tan}^{- 1} x, - 1 < x < 1

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