Question

Integrate the following with respect to x :

`tan x sqrt(sec x)`

Answer 1

`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"`