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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=28`
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