Question
Mon April 11, 2011 By: Rohit Varshney
 

Find all function f(x) defined on {-pi/2,pi/2} with real values and has a primitive F(x) such that f(x) +cos x .F(x)=sin2x/(1+sinx)^2 . Find f(x).

Expert Reply
Mon May 09, 2011
We have:
f(x) = F'(x)          ... (1)
 
The given relation is:
f(x) +cos x .F(x)=sin2x/(1+sinx)2
Multiply the above equation with esinx,
esinxf(x) +esinxcos x .F(x)=esinxsin2x/(1+sinx)2
[esinxF(x)]' = 2 sinx cosx esinx/(1+sinx)2      [Using (1)]
[esinxF(x)]' = 2 [esinx/(1+sinx)]'
Integrating, we get
esinxF(x) = 2 esinx/(1+sinx) + A, where A is a constant
F(x) = 2 /(1+sinx) + Ae-sinx
 
Now, from (1), we have f(x) = F'(x)
So, now differentiate F(x) with respect to x to get the function f(x).
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