AP EAMCET202319 May 2023Morning ShiftMathematicsDefinite IntegrationActual
Consider the following statements (A) and (B) (A) _a^b d d x (f(x)) d x= d d x _a^b(f(x)) d x (B) d d x ( f(x) d x )=f(x)+C Which one of the following is true?
Options
- AOnly (A) is true
- BOnly (B) is true
- CBoth ( A ) and ( B ) are true
- DBoth (A) and (B) are false
Correct answer
D. Both (A) and (B) are false
Step-by-step solution
_a^b d d x (f(x)) d x= _a^b d(f(x))=f(b)-f(a) ...(i) and d d x _a^b f(x) d x= d d x [ ~A constant ]=0 ...(ii) _a^b f(x) d x . will necessarily give a constant value Hence _a^b d d x (f(x)) d x d d x _a^b f(x) d x Therefore statement ' A ' is false. Now consider d d x ( f(x) d x )= d d x (g(x)+C₁ ) where f(x) d x=g(x)+C₁ aligned & d d x f(x) d x=g^ ( x )+0 d d x ( constant )=0 . & =g^ (x) aligned Clearly statement ' B ' is also false.