If `int_a^b(f(x)-3x)dx=a^2-b^2` then the value of `f(pi/6)` is ___

Options

A. `(pi)/(6)`
B. `(2pi)/(3)`
C. `(pi)/(2)`
D. `(pi)/(2)`

Answer

A

Explanation

We have,
`overset(b)underset(a)int {f(x)-3x}dx=a^(2)-b^(2)`
`rArr overset(b)underset(a)int f(x)dx-3 overset(b)underset(a)int x dx=a^(2)-b^(2)`
`rArr overset(b)underset(a)int f(x)dx-(3)/(2)(b^(2)-a^(2))=a^(2)-b^(2)`
`rArr overset(b)underset(a)int f(x)dx=(1)/(2)(b^(2)-a^(2))`
`rArr f(x)=x " "[:'overset(b)underset(a)int x dx=(1)/(2)(b^(2)-a^(2))]`
`rArr f((pi)/(6))=(pi)/(6)`