Thursday, July 25, 2013

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

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.

No comments:

Post a Comment

What accomplishments did Bill Clinton have as president?

Of course, Bill Clinton's presidency will be most clearly remembered for the fact that he was only the second president ever...