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Question
Integrate the following with respect to x :
`tan x sqrt(sec x)`
Solution
`int tan x sqrt(sec x) "d"x = int sinx/cosx * sqrt(sec x) * "d"x`
= `int sin x * sec x sqrt(secx) * "d"x`
= `int sin x * (sec x)^(3/2) * "d"x`
= `int sinx/(cos x)^(3/2) * "d"x`
Put cos x = u
– sin x dx = du
sin x dx = – du
`int tan x sqrt(sec x) "d"x = int (- "du")/"u"^(3/2)`
= `- int "u"^(- 3/2) "du"`
= ` - ("u"^(- 3/2 + 1))/(- 3/2 + 1) + "c"`
= `- ("u"^(- 1/2))/(- 1/2) + "c"`
= `2 "u"^(- 1/2) + "c"`
= `2(cos x)^(- 1/2) + "c"`
= `2 1/(cos x)^(1/2) + "c"`
= `2(sec x)^(1/2) + "c"`
`int tan x sqrt(sec x) "d"x = 2 sqrt(sec x) + "c"`
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