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