Match List-I with List-II
[ \( c \) is an arbitrary constant ]
| List-I | List-II | ||
| (A) | \( \int\displaystyle\frac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\,dx \) | (I) | \( \tan x-x+c \) |
| (B) | \( \int\displaystyle\frac{\sec^2x}{\mathrm{cosec}^2x}\,dx \) | (II) | \( 2\tan x-3\sec x+c \) |
| (C) | \( \int\sec x\,(\sec x+\tan x)\,dx \) | (III) | \( \tan x+\cot x+c \) |
| (D) | \( \int\displaystyle\frac{2-3\sin x}{\cos^2x}\,dx \) | (IV) | \( \tan x+\sec x+c \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
In each case the integrand splits, term by term, into standard forms.
(A) \( \int\frac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\,dx \)
This matches (III) \( \tan x+\cot x+c \).
\( \displaystyle \frac{\sin^2x}{\sin^2x\cos^2x}-\frac{\cos^2x}{\sin^2x\cos^2x}=\sec^2x-\mathrm{cosec}^2x \)
\( \displaystyle \int(\sec^2x-\mathrm{cosec}^2x)\,dx=\tan x-(-\cot x)+c \)
\( \displaystyle =\tan x+\cot x+c \)
(B) \( \int\frac{\sec^2x}{\mathrm{cosec}^2x}\,dx \)
This matches (I) \( \tan x-x+c \).
\( \displaystyle \frac{\sec^2x}{\mathrm{cosec}^2x}=\frac{\sin^2x}{\cos^2x} \)
\( \displaystyle =\tan^2x \)
\( \displaystyle =\sec^2x-1 \)
\( \displaystyle \int(\sec^2x-1)\,dx=\tan x-x+c \)
(C) \( \int\sec x\,(\sec x+\tan x)\,dx \)
This matches (IV) \( \tan x+\sec x+c \).
\( \displaystyle \int(\sec^2x+\sec x\tan x)\,dx=\tan x+\sec x+c \)
(D) \( \int\frac{2-3\sin x}{\cos^2x}\,dx \)
This matches (II) \( 2\tan x-3\sec x+c \).
\( \displaystyle \frac{2}{\cos^2x}-\frac{3\sin x}{\cos^2x}=2\sec^2x-3\sec x\tan x \)
\( \displaystyle \int(2\sec^2x-3\sec x\tan x)\,dx=2\tan x-3\sec x+c \)
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).
NCERT (2026-27 edition):
Integrals
