allery$\int \operatorname{cosec} x\,(\operatorname{cosec} x+\cot x)\,dx$ equals
A$\cot x+\operatorname{cosec} x+C$
B$-\cot x+\operatorname{cosec} x+C$
C$\tan x-\sec x+C$
D$-\cot x-\operatorname{cosec} x+C$
Answer & Solution
Correct answer: D. $-\cot x-\operatorname{cosec} x+C$
1. Expand: $\operatorname{cosec}^2 x+\operatorname{cosec} x\cot x$.
2. $\int \operatorname{cosec}^2 x\,dx=-\cot x$.
3. $\int \operatorname{cosec} x\cot x\,dx=-\operatorname{cosec} x$.
4. Sum: $-\cot x-\operatorname{cosec} x+C$. Option A has both signs flipped.
_Source: syllabus Mathematics Ch 7 "Integrals", p.295_
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