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Let f and g be continuous functions on [ 0 , a] such that `f(x)=f(x)=f(a-x)andg(x)+g(a-x)=4`, then `int_(0)^(a) f(x)g(x) `dx is equal to

A

`(a)/(2)`

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
Let ` l = int_(0)^(a) f(x) . g(x) dx`
` = int_(0)^(a) f(a-x)g(a-x)dx= int_(0)^(a) f(x) {a-g(x)}dx`
` = a int _(0)^(a) f(x) dx - int _(0)^(a) f(x).g(x) dx`
` = a int _(0)^(a) f(x) dx-1 or l = a/2 int_(0)^(2) f(x) dx`
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