Match List-I with List-II

List-I	List-II
Functions	Integrals
---	---
(A) $\int \frac{dx}{x^2 - 4}, x \neq \pm 2$	(I) $\log \vert x + \sqrt{4 + x^2}\vert + C$, where $C$ is an arbitrary constant
(B) $\int \frac{1}{\sqrt{16 - x^2}} dx; \vert x\vert < 4$	(II) $\sin^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant
(C) $\int \frac{1}{16 + x^2} dx$	(III) $\frac{1}{4} \log \left\vert \frac{x-2}{x+2} \right\vert + C$, where $C$ is an arbitrary constant
(D) $\int \frac{1}{\sqrt{4 + x^2}} dx$	(IV) $\frac{1}{4} \tan^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant

Choose the correct answer from the options given below:

Each integral in List-I corresponds to a standard integral formula.

The relevant formulas are:

$\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left|\frac{x-a}{x+a}\right| + C$

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$

$\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$

$\int \frac{dx}{\sqrt{x^2 + a^2}} = \log \left|x + \sqrt{x^2 + a^2}\right| + C$

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(A) $\int \frac{dx}{x^2 - 4}$, where $a = 2$:

$= \frac{1}{2(2)} \log \left|\frac{x-2}{x+2}\right| + C$

$= \frac{1}{4} \log \left|\frac{x-2}{x+2}\right| + C$

$(A) \rightarrow (III)$

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(B) $\int \frac{dx}{\sqrt{16 - x^2}}$, where $a = 4$:

$= \sin^{-1}\left(\frac{x}{4}\right) + C$

$(B) \rightarrow (II)$

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(C) $\int \frac{dx}{16 + x^2}$, where $a = 4$:

$= \frac{1}{4} \tan^{-1}\left(\frac{x}{4}\right) + C$

$(C) \rightarrow (IV)$

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(D) $\int \frac{dx}{\sqrt{4 + x^2}}$, where $a = 2$:

$= \log \left|x + \sqrt{x^2 + 4}\right| + C$

$(D) \rightarrow (I)$

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$(A) \rightarrow (III)$

$(B) \rightarrow (II)$

$(C) \rightarrow (IV)$

$(D) \rightarrow (I)$