If C is an arbitrary constant, then
Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | \( \int\displaystyle \sqrt{x^2+a^2}\,dx \) | I. | \( \displaystyle\frac{x}{2}\sqrt{a^2-x^2}+\displaystyle\frac{a^2}{2}\sin^{-1}\displaystyle\frac{x}{a}+C \) |
| B. | \( \int\displaystyle \sqrt{a^2-x^2}\,dx \) | II. | \( \log\left|x+\sqrt{x^2+a^2}\right|+C \) |
| C. | \( \int\displaystyle \frac{1}{\sqrt{x^2+a^2}}\,dx \) | III. | \( \sin^{-1}\displaystyle\frac{x}{a}+C \) |
| D. | \( \int\displaystyle \frac{1}{\sqrt{a^2-x^2}}\,dx \) | IV. | \( \displaystyle\frac{x}{2}\sqrt{x^2+a^2}+\displaystyle\frac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right|+C \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-IV, B-I, C-II, D-III
These are four standard integrals.
(A) \( \displaystyle\int\sqrt{x^2+a^2}\,dx \)
This matches IV.
\( \displaystyle \int\sqrt{x^2+a^2}\,dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right|+C \)
(B) \( \displaystyle\int\sqrt{a^2-x^2}\,dx \)
This matches I.
\( \displaystyle \int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C \)
(C) \( \displaystyle\int\frac{1}{\sqrt{x^2+a^2}}\,dx \)
This matches II.
\( \displaystyle \int\frac{dx}{\sqrt{x^2+a^2}}=\log\left|x+\sqrt{x^2+a^2}\right|+C \)
(D) \( \displaystyle\int\frac{1}{\sqrt{a^2-x^2}}\,dx \)
This matches III.
\( \displaystyle \int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}+C \)
NCERT (2026-27 edition):
Integrals
