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Question
Evaluate:
`intsqrt(sec x/2 - 1)dx`
Solution
I = `intsqrt(sec x/2 - 1)dx`
= `intsqrt((1 - cos x/2)/(cos x/2)) dx`
= `intsqrt(((1 - cos x/2)(1 + cos x/2))/(cos x/2(1 + cos x/2)))dx`
= `int(sin x/2)/sqrt(cos^2 x/2 + cos x/2)dx`
Let `cos x/2 = t`
`\implies -sin x/2*1/2dx = dt`
`\implies sin x/2*dx = -2dt`
∴ I = `-2int dt/(sqrt(t^2 + t)`
= `-2int dt/sqrt((t + 1/2)^2 - (1/2)^2`
= `-2log_e|(t + 1/2) + sqrt(t^2 + t)| + c`
= `-2log_e|(cos x/2 + 1/2) + sqrt(cos^2 x/2 + cos x/2)| + c`
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