Match List-I with List-II (where \(c\) is an arbitrary constant)
| List-I | List-II | ||
| (A) | \(\int \tan x\, dx\) | (I) | \(\log|\sec x+ \tan x| + c\) |
| (B) | \(\int \cot x\, dx\) | (II) | \(\log|\sec x|+c\) |
| (C) | \(\int \sec x\, dx\) | (III) | \(\log|\sin x| + c\) |
| (D) | \(\int \mathrm{cosec}\, x\, dx\) | (IV) | \(\log|\mathrm{cosec}\,x- \cot x|+c\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (II), (B) - (III), (C) - (I), (D) - (IV)
These are standard integrals. The first two follow from \(\int\frac{g'(x)}{g(x)}\,dx = \log|g(x)| + c\).
(A) \(\int \tan x\, dx\)
This matches (II). With \(\tan x = \frac{\sin x}{\cos x}\) and \((\cos x)' = -\sin x\):
\( \displaystyle \int\tan x\,dx = -\log|\cos x| + c = \log|\sec x| + c \)
(B) \(\int \cot x\, dx\)
This matches (III). With \(\cot x = \frac{\cos x}{\sin x}\) and \((\sin x)' = \cos x\):
\( \displaystyle \int\cot x\,dx = \log|\sin x| + c \)
(C) \(\int \sec x\, dx\)
This matches (I), the standard result \(\log|\sec x + \tan x| + c\).
(D) \(\int \mathrm{cosec}\, x\, dx\)
This matches (IV), the standard result \(\log|\mathrm{cosec}\,x - \cot x| + c\).
So the matching is (A) - (II), (B) - (III), (C) - (I), (D) - (IV).
NCERT (2026-27 edition):
Integrals
