Question

Match each entry in List-I to the correct entries in List-II (where C is the constant of integration).

	List-I		List-II
I	$\int \frac{1}{x^2+a^2} dx$	A	$\ln
II	$\int \frac{1}{\sqrt{a^2-x^2}} dx$	B	$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$
III	$\int \frac{1}{\sqrt{x^2-a^2}} dx$	C	$\ln
IV	$\int \frac{1}{\sqrt{x^2+a^2}} dx$	D	$\sin^{-1}\left(\frac{x}{a}\right)+C$

Answer 1

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

Answer 2

1. The integral $\int \frac{1}{x^2+a^2} dx$ is a standard form which evaluates to $\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$. This matches List-II B.

2. The integral $\int \frac{1}{\sqrt{a^2-x^2}} dx$ is a standard form which evaluates to $\sin^{-1}\left(\frac{x}{a}\right)+C$. This matches List-II D.

3. The integral $\int \frac{1}{\sqrt{x^2-a^2}} dx$ is a standard form which evaluates to $\ln|x+\sqrt{x^2-a^2}|+C$. This matches List-II A.

4. The integral $\int \frac{1}{\sqrt{x^2+a^2}} dx$ is a standard form which evaluates to $\ln|x+\sqrt{x^2+a^2}|+C$. This matches List-II C.