If ϕ(x)=limn →∞ (x^n-x^-11)/(x^n+x^-n), 0<x<1, n ∈N, then ∫sin^-1 x ϕ(x) d x is equal to (A) x s

step 1 So in this question, Phi X is given. So we can write this. Phi X equal, 5x equal limit n tends t

If ϕ(x)=limn →∞ (x^n-x^-11)/(x^n+x^-n), 0<x<1, n ∈N, then...

If $\phi(x)=\lim _{n \rightarrow \infty} \frac{x^{n}-x^{-11}}{x^{n}+x^{-n}}, 0<x<1, n \in N$, then $\int \sin ^{-1} x \phi(x) d x$ is equal to (A) $x \sin ^{-1} x+\sqrt{1-x^{2}}+C$ (B) $-\left(x \sin ^{-1} x+\sqrt{1-x^{2}}\right)+C$ (C) $x \sin ^{-1} x-\sqrt{1-x^{2}}+C$ (D) none of these.

If $\phi(x)=\lim _{n \rightarrow \infty} \frac{x^{n}-x^{-11}}{x^{n}+x^{-n}}, 0<x<1, n \in N$, then
$\int \sin ^{-1} x \phi(x) d x$ is equal to
(A) $x \sin ^{-1} x+\sqrt{1-x^{2}}+C$
(B) $-\left(x \sin ^{-1} x+\sqrt{1-x^{2}}\right)+C$
(C) $x \sin ^{-1} x-\sqrt{1-x^{2}}+C$
(D) none of these.

Key Concepts

Pointwise Convergence

This concept involves analyzing the limit of a sequence of functions at each point separately. When evaluating the limit of functions defined by a parameter (such as n) for each value of x, pointwise convergence examines how the function behaves as the parameter goes to infinity, which can differ across the domain.

Exponential Growth and Decay

In problems involving powers, it is crucial to recognize that if 0 < x < 1, then expressions like x^n decay to zero as n increases, while terms like x^(–n) (or (1/x)^n) grow unboundedly. This contrast is essential in determining the behavior of limits where exponential terms are present in both the numerator and denominator.

Indefinite Integration

Indefinite integration is about finding an antiderivative of a function, which includes an arbitrary constant of integration. It requires identifying a function whose derivative returns the given integrand, often relying on known integration techniques and the properties of the functions involved.

Integration Involving Inverse Trigonometric Functions

Integrals that contain inverse trigonometric functions, like sin?¹x, are common in calculus and typically require special techniques or formulas. Understanding the standard antiderivatives associated with these functions, as well as how they interact with other functions in the integrand, is key for solving such problems.

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