Q.) Find f(x) if f(x)=x+∫xy²+x²yƒ(y)dy. limits of integration - frm 0 to 1.

Dear navroze

f(x)=x+_o∫¹[xy²+x²yƒ(y)]dy

f(x) = x+ x/3  + x²_o∫¹yƒ(y)dy

 _o∫¹yƒ(y)]dy    it is a constant quantity let it is equal to c

so  

f(x) = 4 x/3  + cx²

or f(y) = 4 y/3  + cy²

now  c =_o∫¹yƒ(y)dy

put value of f(y)

  c= _o∫¹y[4 y/3  + cy²]dy

 c = 4/9 + c/4

 3c/4 = 4/9

  c =16/27

so f(x) = 4 x/3  + 16/27 x²

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