The value of `int(dx)/(sqrt(sin(2x+theta)+sintheta))` is

Options

A. `sqrt((tan x+tan theta)sec theta)+c`
B. `sqrt(2 (tan x+tan theta)sec theta)+c`
C. `sqrt(2(sin x+tan theta)sec theta)+c`
D. None of above

Answer

B

Explanation

Let ` l = int ( sec x dx ) /( sqrt ( sin ( 2 x + theta ) + sin theta ) `
` = int ( sec xdx ) /( sqrt ( 2 sin ( x + theta ) cos x ) `
` = (1 ) /( sqrt 2 ) int ( sec^( 3//2 ) x dx ) /( sqrt ( sin x cos theta + sin theta cos x ) `
` = ( 1 ) /(sqrt 2 ) int ( sec^ 2 x dx ) /( sqrt ( tan x cos theta + sin theta ) `
Put ` tan x = t rArr sec ^ 2 x dx = dt `
` therefore I = ( 1 ) /( sqrt 2 ) int ( dt ) /(sqrt ( t cos theta + sin theta ) `
`= ( 2 ) /( sqrt 2 ) ( sqrt ( t cos theta + sin theta ) /( cos theta ) + C )`
` = sqrt ( 2 ( tan x sec theta + tan theta sec theta ) ) + C `