Question

Evaluate \[\int\frac{1-\sin x}{\cos^2x}\,dx.\]

Options

• \[\tan x+\sec x+C\]

• \[-\tan x-\sec x+C\]

• \[\tan x-\sec x+C\]

• \[-\cot x-\operatorname{cosec}x+C\]

Answer 1

Write the integrand as \[\sec^2x-\tan x\sec x.\] Integrating gives \[\tan x-\sec x+C,\] because \[\int\sec^2x\,dx=\tan x\] and \[\int\tan x\sec x\,dx=\sec x.\]