What is f(x) if derivative is f'(x)=sin2x/(sin^2 x-4)?

Expert Answers
justaguide eNotes educator| Certified Educator

We need to find f(x) given that f'(x) = sin 2x /((sin x)^2 - 4)

let ((sin x)^2 - 4) = y

dy = 2*sin x * cos x  dx

=>dy = sin 2x dx

Int [ sin 2x /((sin x)^2 - 4) dx]

=> Int [ 1/y dy]

=> ln |y | + C

substitute y = ((sin x)^2 - 4)

=> ln |((sin x)^2 - 4)| + C

The function f(x)  = ln |((sin x)^2 - 4)| + C

giorgiana1976 | Student

To find the primitive of the given function, we'll have to determine the indefinite integral of sin2x/[(sin x)^2-4]

We notice that if we'll substitute (sin x)^2-4 by t and we'll differentiate both sides, we'll get:

2sinx*cosx dx = dt

We also notice that we may replace the numerator sin 2x, using the identity: sin 2x = 2sinx*cosx

We'll re-write the integral of the function of variable t:

Int sin 2x dx/[(sin x)^2-4] = Int 2sinx*cosx dx/[(sin x)^2-4]

Int 2sinx*cosx dx/[(sin x)^2-4] = Int dt/t

Int dt/t = ln |t| + C

We'll replace t by (sin x)^2-4 and we'll get:

Int sin 2x dx/[(sin x)^2-4] = ln |(sin x)^2-4| + C

The primitive of the given function is F(x) = ln |(sin x)^2-4| + C.