Home
Class 12
MATHS
If for nonzero x ,af(x)+bf(1/x)=1/x-5, w...

If for nonzero `x ,af(x)+bf(1/x)=1/x-5,` where `a!=b` then `int_1^2f(x)dx=`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(a^(2)-b^(2))[a log 2-5a+(7)/(2)b]`
Given , of ` (x)+bf (1//x)=(1)/(x)-5` . . . (i)
Replacing x by `1//x` in Eq . (i) , we get
af `(1//x) +bf (x)=x-5` .. . (ii)
On multiplying Eq . (i) by a and Eqs . (ii) by b , we get
`a^(2)f(x)+abf(1//x)=a((1)/(x)-5)`. . . (iii)
`abf(1//x)+b^(2)f(x)=b(x-5)` ... (iv)
On subtracting Eq . (iv) from Eq . (iii) , we get
`(a^(2)-b^(2)) f(x)=(a)/(x)-bx-5a+5b`
`rArr f(x)=(1)/((a^(2)-b^(2)))((a)/(x)-bx-5a+5b)`
`rArr int_(1)^(2)f(x)dx=(1)/((a^(2)-b^(2)))int_(1)^(2)((a)/(x)-bx-5a+5b)dx`
`=(1)/((a^(2)-b^(2)))[alog|x|-(b)/(2)x^(2)-5(a-b)x]_(1)^(2)`
`=(1)/((a^(2)-b^(2)))[alog2-2b-10(a-b)-alog1+(b)/(2)+5(a-b)]`
`=(1)/((a^(2)-b^(2)))[alog 2-5a+(7)/(2)b]`
Promotional Banner

Similar Questions

Explore conceptually related problems

Evaluate: int1/(x^2-x+1)dx

Evaluate: int1/(2x^2+x-1)dx

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dxdot

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

If f(x) is a function satisfying f(1/x)+x^2f(x)=0 for all nonzero x , then evaluate int_(sintheta)^(cos e ctheta)f(x)dx

If g(x) is the inverse of f(x) and f(x) has domain x in [1,5] , where f(1)=2 and f(5) = 10 then the values of int_1^5 f(x)dx+int_2^10 g(y) dy equals

Let f(x) = x-[x] , for every real number x, where [x] is integral part of x. Then int_(-1) ^1 f(x) dx is

A continuous real function f satisfies f(2x)=3(f(x)AAx in RdotIfint_0^1f(x)dx=1, then find the value of int_1^2f(x)dx

A continuous real function f satisfies f(2x)=3(f(x)AAx in RdotIfint_0^1f(x)dx=1, then find the value of int_1^2f(x)dx