Question

Let `f:(0,oo)to R` be a continuous function such that `F(x)=int_(0)^(x^(2)) tf(t)dt. "If "F(x^(2))=x^(4)+x^(5),"then "sum_(r=1)^(12) f(r^(2))=`
A. 216
B. 219
C. 222
D. 225

Answer 1

Correct Answer - B
It is given that f is continuous. Therefore, the integral function f(x) is differentiable. Also,
`F(x)=underset(0)overset(x)int t(f(t)dt`
`F(x^(2))=underset(0)overset(x^(2))int t(f(t)dt`
`x^(4)+x^(5)=underset(0)overset(x^(2))int t(f(t)dt`
Differentiating with respect to x, we get
`4x^(3)+5x^(4)=(2x)x^(2)f(x^(2))`
`Rightarrow f(x^(2))=2+(5)/(2)x`
`Rightarrow f(r^(2))=2+(5)/(2)r`
`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`