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AP EAMCET202315 May 2023Morning ShiftMathematicsDefinite IntegrationActual

If f(x)= x^3+5 12+x and _ -5 ^5 f(x) d x= ₀^5(f(x)+g(x)) d x , then g(x)=

Options

  1. A5-x^3 12-x
  2. B- ( 5+ x ^3 12+x )
  3. C- x ^3+5 12+ x
  4. D5+x^3 12-x

Correct answer

A. 5-x^3 12-x

Step-by-step solution

Let I= _ -5 ^5 x^3+5 12+x d x...(1) I= _ -5 ^5 (5-5-x)^3+5 12+(5-5-x) d x...(By Property)...(2) Adding equation (1) & (2) 2 I= _ -5 ^5 ( x^3+5 12+x + -x^3+5 12-x ) d x Since f(-x)=f(x) aligned & Hence 2 I=2 ₀^5 ( x^3+5 12+x + 5-x^3 12-x ) d x & _ -5 ^5 x^3+5 12+x d x= ₀^5 ( x^3+5 12+x + 5-x^3 12-x ) d x & By comparision, g(x)= 5-x^3 12-x . aligned

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