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216219222225

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It is given that f is continuous. Therefore, the integral function f(x) is differentiable. Also, <br> `F(x)=underset(0)overset(x)int t(f(t)dt` <br> `F(x^(2))=underset(0)overset(x^(2))int t(f(t)dt` <br> `x^(4)+x^(5)=underset(0)overset(x^(2))int t(f(t)dt` <br> Differentiating with respect to x, we get <br> `4x^(3)+5x^(4)=(2x)x^(2)f(x^(2))` <br> `Rightarrow f(x^(2))=2+(5)/(2)x` <br> `Rightarrow f(r^(2))=2+(5)/(2)r` <br> `Rightarrow underset(r=1)overset(12)sum f(r^(2))=underset(r=1)overset(12)sum (2+(5)/(2)r)=24+(5)/(2)xx78=219`