Question

Whats is the integral of sinx/sq.root of 4cos square x-1?

Answer 1

Hey there !!!!

$\int\ sinx/ \sqrt{4cos^2x-1} \, dx$----Equation 1


Let cosx=t

Differentiating with wrt to "t"

   -sinxdx=dt

Substituting -sinx.dx=dt and cosx=t in equation 1


=$\int\ (-1/ \sqrt{4t^2-1^2} ) \, dt$


$\int\ -1/ \sqrt{(2t)^2-1^2} \, dt$---Equation 2

Equation 2 is of the form :

$\int\ 1/ \sqrt{x^2-a^2} \, dx$=cosh⁻¹(x/a)+c



$\int\ -1/ \sqrt{(2t)^2-1^2} \, dt$ =-cosh⁻¹(2t)+c

                                                              
                                                             =-cosh⁻¹(2cosx)+c