If a`f(x)`+b`f((1)/(x))=(1)/(x)-5,x ne 0,a ne b`, then `overset(2)underset(1)int f(x)dx` equals

Options

A. `(("In"2-5)a+(13)/(2)b)/(a^(2)-b^(2))`
B. `(("In"2-5)a+(7b)/(2))/(a^(2)-b^(2))`
C. `((5-"In"2)a+(7b)/(2))/(a^(2)-b^(2))`
D. none of these

Answer

B

Explanation

We have,
`af(x)+bf((1)/(x))=(1)/(x)-5 " "`......(i)
`rArrbf(x)+a(f)((1)/(x))=x-5" "["Replacing x by (1)/(x)]`.......(ii)
Solving these equations for f(x), we get
f(x)`=(1)/(a^(2)-b^(2)){(a)/(x)-bx-5a+5b}`
`f(x)dx=(1)/(a^(2)-b^(2))[a " In "x-b-(x^(2))/(2)-5(a-b)x]_(1)^(2)`
`rArr overset(2)underset(1)intf(x)dx=(1)/(a^(2)-b^(2))[a" In "2-(3b)/(2)-5(a-b)]`
`rArr overset(2)underset(1)int dx=(1)/(a^(2)-b^(2))[("In "2-5)a+(7b)/(2)]`