What is the best substitution to make to evaluate the integral of the quotient of cosine of 2 times x and the square root of the quantity 5 minus 2 times the sine of 2 times x, dx?

Question

What is the best substitution to make to evaluate ?

 u = sin(2x) 

u = cos(2x) 

u = 2x 

u = 5 − 2sin(2x)

Answer 1

∫[cos(2x) / √(5 - 2sin(2x)) dx

Let u = 5 - 2sin(2x). Then, du = -4cos(2x)dx. So, cos(2x)dx = (-1/4)du

The integral can then be rewritten as (-1/4)∫ [du / √u] = -(1/2)√u + C = -(1/2)√[5 - 2sin(2x)] + C