If Φ ( x ) = lim n → ∞ x n − x − n x n + x − n , 0 < x < 1 , ϵ N , then ∫ ( s i n − 1 x ) ( Φ ( x ) ) d x is equal to Options: A . &nbsp x s i n − 1 ( x ) + √ 1 − x 2 + C B . &nbsp − ( x s i n − 1 ( x ) + √ 1 − x 2 ) C . &nbsp x s i n − 1 x + √ 1 − x 2 + C D . &nbsp None of these

Answer: Option A : A We have ϕ ( x ) = lim n → ∞ x 2 n − 1 x 2 n + 1 = 1 , 0 < x < 1 ∴ ∫ s i n − 1 x . ϕ ( x ) d x = ∫ s i n − 1 x d x = [ x s i n − 1 x + √ 1 − x 2 ] + c

If Φ(x)=limn→∞xn−x−nxn+x−n,0<x<

Discussion Forum : Indefinite Integration

Question -

If $Φ (x) = {lim}_{n \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}}, 0 < x < 1, ϵ N$ , then $\int (s i n^{- 1} x) (Φ (x)) d x$ is equal to

Options:

A .

x s i n^{- 1} (x) + \sqrt{1 - x^{2}} + C

B .

- (x s i n^{- 1} (x) + \sqrt{1 - x^{2}})

C .

x s i n^{- 1 x + \sqrt{1 - x^{2}} + C}

D . None of these

Answer: Option A
:
A
We have

ϕ (x) = {lim}_{n \to \infty} \frac{x^{2 n} - 1}{x^{2 n} + 1} = 1, 0 < x < 1 ∴ \int s i n^{- 1} x . ϕ (x) d x = \int s i n^{- 1} x d x = [x s i n^{- 1} x + \sqrt{1 - x^{2}}] + c

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Discussion Forum : Indefinite Integration

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