Match List-I with List-II

List-I	List-II
Integral	Solution: C is an arbitrary constant
---	---
(A) $\int \frac{dx}{x^2 + 25}$	(I) $\frac{1}{10} \log \left\vert \frac{5 + x}{5 - x} \right\vert + C$
(B) $\int \frac{dx}{x^2 - 25}$	(II) $\log \vert x + \sqrt{x^2 - 25}\vert + C$
(C) $\int \frac{dx}{25 - x^2}$	(III) $\frac{1}{5} \tan^{-1} \left( \frac{x}{5} \right) + C$
(D) $\int \frac{dx}{\sqrt{x^2 - 25}}$	(IV) $\frac{1}{10} \log \left\vert \frac{x - 5}{x + 5} \right\vert + C$

Choose the correct answer from the options given below:

Standard integral formulas with $a = 5$ (since $25 = 5^2$):

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

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

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

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

(A) $\int \frac{dx}{x^2 + 25}$, with $a = 5$:

$= \frac{1}{5}\tan^{-1}\left(\frac{x}{5}\right) + C \quad \longrightarrow \text{(III)}$

(B) $\int \frac{dx}{x^2 - 25}$, with $a = 5$:

$= \frac{1}{2(5)}\log\left|\frac{x - 5}{x + 5}\right| + C$

$= \frac{1}{10}\log\left|\frac{x-5}{x+5}\right| + C \quad \longrightarrow \text{(IV)}$

(C) $\int \frac{dx}{25 - x^2}$, with $a = 5$:

$= \frac{1}{2(5)}\log\left|\frac{5 + x}{5 - x}\right| + C$

$= \frac{1}{10}\log\left|\frac{5+x}{5-x}\right| + C \quad \longrightarrow \text{(I)}$

Note: In (B), $x^2$ comes first → $(x-a)$ on top. In (C), $a^2$ comes first → $(a+x)$ on top.

(D) $\int \frac{dx}{\sqrt{x^2 - 25}}$, with $a = 5$:

$= \log\left|x + \sqrt{x^2 - 25}\right| + C \quad \longrightarrow \text{(II)}$

Final matching:

(A) → (III)

(B) → (IV)

(C) → (I)

(D) → (II)

Therefore, the correct answer is Option 2.