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Let k(x) be a continuous function satisfying the equation ₀^ x^3 k(t) d t=x^ 1+x^2 , find the value of 3 k(1) .

Correct answer

2

Step-by-step solution

By Leibnitz Theorem, d d x ₀^ x^3 k(t) d t= d d x (x^ 1+x^2 ) k (x^3 ) 3 x^2-0= (1+x^2 ) x^ x^2 +x^ 1+x^2 ( x) (2 x) Put x=1 k(1) 3=2 1+1 0 2 k(1)= 2 3 3 k(1)=2

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