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AP EAMCET20227 Jul 2022Morning ShiftMathematicsIndefinite IntegrationActual

If f(x) is anti-derivative of g(x) and f(x) g(x) (1+f^2(x) ) d x=F(x) , then F(x)=

Options

  1. A. (1+f^2 x ) )^2 4 +C
  2. B. (1+f^2 x ) )^2 2 +C
  3. Cf^2(x) g(x) 4 +C
  4. Dg^2(x) f(x) 4 +C

Correct answer

A. . (1+f^2 x ) )^2 4 +C

Step-by-step solution

Given, f(x)= g(x) d x f^ (x)=g(x)F(x)= f(x) g(x) (1+f^2(x) ) d x On putting 1+f^2(x)=t 2 f(x): f^ (x) d x=d t f(x) g(x) d x= d t 2 [ f^ (x)=g(x) ]f(x)= t d t 2 = 1 2 t^2 2 +C= (1+f^2(x) )^2 4 +C

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