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Question
`int (dx)/(sin^2 x cos^2 x)` equals:
Options
tan x + cot x + C
tan x - cot x + C
tan x cot x + C
tan x - cot 2x + C
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Solution
tan x - cot x + C
Explanation:
Let `I = int dx/(sin^2 cos^2 x)`
`= int (sin^2 x cos^2 x)/(sin^2 x cos^2 x) dx`
`= int ((sin^2 x)/(sin^2 x cos^2 x) + (cos^2 x)/(sin^2 x cos^2 x)) dx`
`= int (1/(cos^2 x) + 1/(sin^2 x)) dx`
`= int (sec^2 x + cosec^2 x) dx`
`= int sec^2 dx + int cosec^2 x dx`
= tan x - cot x + C
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