Home
Class 12
MATHS
If g(x) is the inverse of f(x) and f(x...

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

A

72

B

56

C

36

D

48

Text Solution

Verified by Experts

The correct Answer is:
D
`y=f(x)`
`rArr" "x=f^(-1)(y)=g(y)`
`" "dy=f'(x)dx`
where y is 2 then x = 1 and y is 10 then x = 5
`therefore" "I=int_(1)^(5)f(x)dx+int_(2)^(10)g(y)dy`
`" "=int_(1)^(5)f(x)dx+int_(1)^(5)xf'(x)dx`
`therefore" "I=int_(1)^(5)(f(x)+xf'(x))dx`
`" "=xf(x)|_(1)^(5)=5f(5)-f(1)=5.10-2=48`
Promotional Banner

Similar Questions

Explore conceptually related problems

If f(-x)=-f(x) , then the value of int_(-a)^af(x)dx is equal to

If g (x) is the inverse of f (x) and f(x)=(1)/(1+x^(3)) , then find g(x) .

If (d)/(dx)f(x)=g(x) , then the value of int_(a)^(b)f(x)g(x)dx is -

If int_[x]^(x+1) f(t)dt =[x] then th value of int_(-2)^4 f(x) dx is equal

If int_(-1)^4 f(x) dx =4 and int_2^4[3-f(x)]dx=7 then the value of int_(-1)^2 f(x) dx is

2f(1/x) - f(x) = 5x , find the value of f(x + 1/x).

Let y=f(x)=4x^(3)+2x-6 , then the value of int_(0)^(2)f(x)dx+int_(0)^(30)f^(-1)(y)dy is equal to _________.

If f(0)=1,f(2)=3,f^'(2)=5 ,then find the value of int_0^1xf^('')(2x)dx

If for all values of x and y , f(x+y) =f(x) f(y) and f(5) =2, f'(0)=3, then the value of f'(5) is-

It g(x) is the inverse of f(x) and f(X) = (1+x^3)^-1 , show that g'(x) = 1/(1 + {g(x)}^3) .