Match List-I with List-II
| List-I | List-II |
| Indefinite integral | Solution (where ‘c’ is an arbitrary constant) |
| (A) \( \int\sqrt{x^2-a^2}\,dx \) | (I) \( \displaystyle\frac{1}{a}\tan^{-1}\displaystyle\frac{x}{a}+c \) |
| (B) \( \int\displaystyle\frac{1}{x^2-a^2}\,dx \) | (II) \( \displaystyle\frac{x}{2}\sqrt{x^2+a^2}+\displaystyle\frac{a^2}{2}\,log\left[x+\sqrt{x^2+a^2}\right]+c \) |
| (C) \( \int\displaystyle\frac{1}{x^2+a^2}\,dx \) | (III) \( \displaystyle\frac{x}{2}\sqrt{x^2-a^2}-\displaystyle\frac{a^2}{2}\,log\left[x+\sqrt{x^2-a^2}\right]+c \) |
| (D) \( \int\sqrt{x^2+a^2}\,dx \) | (IV) \( \displaystyle\frac{1}{2a}\log\left|\displaystyle\frac{x-a}{x+a}\right|+c \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
These are four standard integrals.
(A) \( \int\sqrt{x^2-a^2}\,dx \)
This matches (III).
\( \displaystyle \int\sqrt{x^2-a^2}\,dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c \)
(B) \( \int\displaystyle\frac{1}{x^2-a^2}\,dx \)
This matches (IV).
\( \displaystyle \int\frac{dx}{x^2-a^2}=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+c \)
(C) \( \int\displaystyle\frac{1}{x^2+a^2}\,dx \)
This matches (I).
\( \displaystyle \int\frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}+c \)
(D) \( \int\sqrt{x^2+a^2}\,dx \)
This matches (II).
\( \displaystyle \int\sqrt{x^2+a^2}\,dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right|+c \)
So the matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II).
NCERT (2026-27 edition):
Integrals
