Integrate sqrt(tanx)sec^4(x)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to determine `int sqrt(tan x)sec^4x dx`

Let `y = sqrt(tan x)` , `dy/dx = (1/2)*(sec^2x)/sqrt(tan x)`

=> `2*sqrt(tan x) dy = sec^2x*dx`

=> `2*y dy = sec^2 x dx`

`int sqrt(tan x)sec^4x dx`

use the relation `sec^2x = 1 + tan^2x`

=> `int y*(1 + y^4)*2y dy`

=> `int 2y^2 + 2y^6 dy`

=> `2*y^3/3 + 2*y^7/7`

=> `(14y^3 + 6y^7)/21`

=> `(y(14y^2 + 6y^6))/21`

substitute y = `sqrt(tan x)`

=> `((sqrt tan x)*(14*tan x + 6*tan^3 x))/21`

The required integral is `((sqrt tan x)(14*tan x + 6*tan^3 x))/21 + C`

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