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.
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