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If A(x+y)=A(x)A(y) and A(0) ne 0 and B(...

If `A(x+y)=A(x)A(y) and A(0) ne 0 and B(x)=(A(x))/(1+(A(x))^(2))`, then

A

`int_(-2010)^(2010)B(x)dx=int_(0)^(2011)B(x)dx`

B

`int_(-2010)^(2011)B(x)dx=int_(0)^(2010)B(x)dx+int_(0)^(2011)B(x)dx`

C

`int_(-2010)^(2011)B(x)dx=0`

D

`int_(-2010)^(2010)(2B(-x)-B(x))dx=2int_(0)^(2010)B(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
B:D
`A(x+y)=A(x)A(y)`
`rArr" "A(0+0)=A(0)A(0)`
`rArr" "A(0)=1`
Put `y=-x,` we get
`A(0)=A(x)A(-x)" (i)"`
`B(-x)=(A(-x))/(1+(A(-x))^(2))`
`=((1)/(A(x)))/(1+(1)/((A(x))^(2)))`
`=(A(x))/(1+(A(x))^(2))`
= B(x)
Thus, B(x) is even.
`int_(-2010)^(2011)B(x)dx=int_(-2010)^(2010)B(x)dx+int_(2010)^(2011)B(x)dx`
`=2int_(0)^(2010)B(x)dx+int_(2010)^(2011)B(x)dx`
`=int_(0)^(2010)B(x)dx+int_(0)^(2011)B(x)dx`
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