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Question
Find the integrals of the function:
`1/(sin xcos^3 x)`
Solution
Let `I = int 1/(sin x cos^3 x) dx`
`= int (sin^2 x + cos^2 x)/(sin x cos^3 x) dx`
`= int ((sin^2 x)/(sin x cos^3 x) + (cos^2 x)/(sin x cos^3 x)) dx`
`= int ((sin x)/(cos 3 x) + (cos x)/(sin x cos^2 x)) dx`
`= int (tan x sec^2 x + (sec^2 x)/(tan x)) dx`
`= int (tan x + 1/(tan x)) sec^2 x dx`
Put tan x = t
⇒ sec2 x dx = dt
∴ `I = int (t + 1/t) dt `
`= t^2/2 + log |t| + C`
`log |tan x| + 1/2 tan^2 x + C`
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