If the graph of the anti derivative g(x) of f(x)= ( )+( x)⁻² passes through (e, 2023-e) and the term independent of x in g(x) is k , then the sum of all the digits of k is

Step-by-step solution

Given : f(x)= ( x)+( x)⁻² Anti-derivative of f(x)= ( ( x)+( x)⁻² ) d x ...(i) Let t= x x=e^t d x=e^t d t Using above values in eqn. (i), we get : Anti-derivative of f(x)= e^t ( t+t⁻² ) d t aligned & g(x)= e^t ( t+t⁻¹-t⁻¹+t⁻² ) d t & g(x)= e^t ( t+t⁻¹ ) d t+ e^t (-t⁻¹+t⁻² ) d t & =e^t t-e^t t⁻¹+C & =e^t ( t-t⁻¹ )+C aligned g(x)=e^ ( x) ( ( x)-( x)⁻¹ )+C ...(i) Equation (i) passes through (e, 2023-e) Eqn. (i) reduces to : 2023-e=e^ ( e) ( ( e)-( e)⁻¹ )+C aligned & 2023-e=e( (1)-1)+C & 2023-e=-e+C C=2023 aligned Putti