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Let f and g be continuous fuctions on [0...

Let f and g be continuous fuctions on [0, a] such that `f(x)=f(a-x)" and "g(x)+g(a-x)=4 " then " underset0oversetaintf(x)g(x)dx` is equal to

A

`4int_(0)^(a)f(x)`dx

B

`int_(0)^(a)f(x)`dx

C

`2int_(0)^(a)f(x)`dx

D

`-3int_(0)^(a)f(x)`dx

Text Solution

Verified by Experts

The correct Answer is:
C
Let `I=int_(0)^(a)f(x)g(x)dx` . . . (i)
`=int_(0)^(a)f(a-x)g(a-x)dx`
`[:'int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx]`
`rArrI=int_(0)^(a)f(x)[4-g(x)]dx`
`[:' f(x)=f(a-x)andg(x) +g (a-x)=4]`
`=int_(0)^(a)4f(x)dx - int_(0)^(a)f(x)g(x)dx`
`rArrI=4int_(0)^(a)f(x)dx -I` [ from Eq . (i)]
`2I=4int_(0)^(a)f(x)dxrArrI=2int_(0)^(a)(x) dx`.
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