allery$\int \dfrac{1-\sin x}{\cos^2 x}\,dx$ equals
A$\tan x+\sec x+C$
B$\sec x-\tan x+C$
C$\tan x-\sec x+C$
D$\cot x-\operatorname{cosec} x+C$
Answer & Solution
Correct answer: C. $\tan x-\sec x+C$
1. Split: $\dfrac{1}{\cos^2 x}-\dfrac{\sin x}{\cos^2 x}=\sec^2 x-\tan x\sec x$.
2. $\int \sec^2 x\,dx=\tan x$.
3. $\int \sec x\tan x\,dx=\sec x$, subtracted gives $-\sec x$.
4. Result: $\tan x-\sec x+C$. Option A wrongly adds $\sec x$.
_Source: syllabus Mathematics Ch 7 "Integrals", p.295_
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