Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(IV), (C)-(I), (D)-(II)

The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

Explanation and Matching:

(A) $\int \frac{dx}{x^{2}+25}$

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

Here $a=5$ ⇒ $\frac{1}{5}\tan^{-1}\left(\frac{x}{5}\right)+C$ → (III)

(B) $\int \frac{dx}{\sqrt{x^{2}-25}}$

Standard form: $\int \frac{dx}{\sqrt{x^{2}-a^{2}}} = \log|x+\sqrt{x^{2}-a^{2}}|+C$

Here $a=5$ ⇒ $\log|x+\sqrt{x^{2}-25}|+C$ → (II)

(C) $\int \frac{dx}{25-x^{2}}$

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

Here $a=5$ ⇒ $\frac{1}{10}\log\left|\frac{5+x}{5-x}\right|+C$ → (I)

(D) $\int \frac{dx}{x^{2}-25}$

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

Here $a=5$ ⇒ $\frac{1}{10}\log\left|\frac{5-x}{5+x}\right|+C$ → (IV)

Final Matching:

(A) → (III), (B) → (II), (C) → (I), (D) → (IV)