Match List-I with List-II (C is a constant of integration)
| List-I | List-II | ||
| (A) | \(\int\tan x\sec^2x\,dx\) | (I) | \(\displaystyle -\frac{1}{1+\tan x}+C\) |
| (B) | \(\int\displaystyle\frac{\sec^2x}{\tan x}\,dx\) | (II) | \(\displaystyle \frac{1}{2}\tan^2x+C\) |
| (C) | \(\int\cot x\,\mathrm{cosec}^2x\,dx\) | (III) | \(\log\left|\tan x\right|+C\) |
| (D) | \(\int\displaystyle\frac{\sec^2x}{\left(1+\tan x\right)^2}\,dx\) | (IV) | \(\displaystyle -\frac{1}{2}\cot^2x+C\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
In each integral the factor \(\sec^2x\) or \(\mathrm{cosec}^2x\) is (up to sign) the derivative of \(\tan x\) or \(\cot x\), so a substitution reduces it to a power of t.
(A) \(\int\tan x\sec^2x\,dx\)
This matches (II) \(\frac{1}{2}\tan^2x+C\).
\( \displaystyle t = \tan x:\quad \int t\,dt = \frac{t^2}{2} \)
(B) \(\int\frac{\sec^2x}{\tan x}\,dx\)
This matches (III) \(\log\left|\tan x\right|+C\).
\( \displaystyle t = \tan x:\quad \int\frac{dt}{t} = \log\left|t\right| \)
(C) \(\int\cot x\,\mathrm{cosec}^2x\,dx\)
This matches (IV) \(-\frac{1}{2}\cot^2x+C\).
\( \displaystyle t = \cot x,\ dt = -\mathrm{cosec}^2x\,dx:\quad -\int t\,dt = -\frac{t^2}{2} \)
(D) \(\int\frac{\sec^2x}{\left(1+\tan x\right)^2}\,dx\)
This matches (I) \(-\frac{1}{1+\tan x}+C\).
\( \displaystyle t = 1+\tan x:\quad \int\frac{dt}{t^2} = -\frac{1}{t} \)