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Let f: Rveca n dg: RvecR be continuous f...

Let `f: Rveca n dg: RvecR` be continuous function. Then the value of the integral `int_(-pi/2)^(pi/2)[f(x)+f(-x)][g(x)-g(-x)]dxi s` `pi` (b) 1 (c) `-1` (d) 0

A

`pi`

B

1

C

`-1`

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int_(-pi//2)^(pi//2)[f(x)+f(-x)][g(x)-g(-x)]dx`
Let `phi(x)=[f(x)+f(-x)][g(x)-g(-x)]`
`rArrphi(-x)=[f(-x)+f(x)](g(-x)-g(x)]`
`rArrphi(-x)=-phi(x)`
`rArrphi(x)` is an odd function.
`:. int_(-pi//2)^(pi//2)phi(x)dx=0`
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