If f(x)=limn →∞ (x^2 n+x^-2 n)/(x^2 n-x^-2 n),(x>1) then ∫(x ·f(x) ·log(x+√(1+x^2)))/(√(1+x^2))

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If f(x)=limn →∞ (x^2 n+x^-2 n)/(x^2 n-x^-2 n),(x>1) then ∫(x...

If $f(x)=\lim _{n \rightarrow \infty} \frac{x^{2 n}+x^{-2 n}}{x^{2 n}-x^{-2 n}},(x>1)$ then $\int \frac{x \cdot f(x) \cdot \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x$ is (a) $\sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)+x+C$ (b) $\sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)-x+C$ (c) $\log \left(x+\sqrt{1+x^{2}}\right)+x \sqrt{1+x^{2}}+C$ (d) $\log \left(x+\sqrt{1+x^{2}}\right)-x-\sqrt{1+x^{2}}+C$

If $f(x)=\lim _{n \rightarrow \infty} \frac{x^{2 n}+x^{-2 n}}{x^{2 n}-x^{-2 n}},(x>1)$ then $\int \frac{x \cdot f(x) \cdot \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x$ is
(a) $\sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)+x+C$
(b) $\sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)-x+C$
(c) $\log \left(x+\sqrt{1+x^{2}}\right)+x \sqrt{1+x^{2}}+C$
(d) $\log \left(x+\sqrt{1+x^{2}}\right)-x-\sqrt{1+x^{2}}+C$

Key Concepts

Limit of Exponential Functions

This concept involves evaluating the behavior of functions that contain terms with exponents becoming arbitrarily large. In cases where the base of the exponent is greater than one, the term with the positive exponent dominates, while terms with negative or smaller exponents become negligible. Recognizing the dominant behavior in limits is crucial to simplifying expressions that involve extreme exponentiation.

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation. It is used to integrate products of functions by choosing parts of the integrand to differentiate and others to integrate, thereby transforming the original integral into one that is easier to evaluate. This method is particularly useful when one factor of the product simplifies upon differentiation.

Chain Rule and Differentiation of Composite Functions

The chain rule is fundamental in differentiating composite functions. When a function is composed of multiple nested functions, the chain rule enables the computation of the derivative by differentiating the outer function and multiplying it by the derivative of the inner function. This technique is essential in verifying antiderivatives and in working with functions that involve compositions, such as logarithms of functions involving square roots.

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as the inverse hyperbolic sine, provide alternative representations for certain logarithmic expressions. For example, the formula asinh(x) = ln(x + ?(1 + x²)) connects inverse hyperbolic functions to logarithm expressions with square roots. This relationship is useful for transforming and simplifying integrals and derivatives that contain such logarithmic forms.

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