If `int_(2x^(2))^(x^(3)) (In x)f(t) dt=x^(2)-2x+5`, then f(8)=

Options

A. `(1)/("In"2)`
B. `(1)/(2"In"2)`
C. `(1)/(3"In"2)`
D. `(1)/(4"In"2)`

Answer

B

Explanation

We have,
`underset(2x^(2))overset(x^(3))int ("In "x)f(t) dt=x^(2)-2x+5`
`rArr (d)/(dx){underset(2x^(2))overset(x^(3))int ("In "x)f(t)dt}=(d)/(dx)(x^(2)-2x+5)`
`rArr underset(2x^(2))overset(x^(3))int(1)/(x)f(t) dt+3x^(2)("In "x)f(x^(3))-4x("In "x)f(2x^(2))=2x-2`
Replacing x=2, we obtain
12(In 2)f(8)-8(In 2)f(8)=2
`rArr f(8)=(1)/(2"In"2)`