The value of
`int_(-1)^(1) ((logx+sqrt(1+x^(2))))/(x+log(x+sqrt(1+x^(2))))f(x) dx-int_(-1)^(1) (log(x sqrt(1+x^(2))))/(x+log(x+logsqrt(1+x^(2))))f(-x)dx`,

Options

A. 0
B. `2underset(0)overset(1)int (log(x+sqrt(1+x^(2))))/(x+log(x+sqrt(1+x^(2)))){f(x) -f(-x)}dx`
C. 2f(x)
D. none of these

Answer

A

Explanation

Let`g(x)= (log(x+sqrt(1+x^(2))))/(x+log(x+sqrt(1+x^(2))))`and let I be the value of the given expression. Then,
`I=underset(-1)overset(1)int g(x)f(x)dx-underset(-1)overset(1)int g(x)f(-x)dx`
`rArr I=underset(-1)overset(1)int g(x){f(x)-f(-x)}dx`
`rArr I=underset(-1)overset(1)int g(x) h(x) dx`,where h(x)=f(x)-f(-x)
We observe that g(x) and h(x) are even and odd functions respectively. Therefore, g(x)h(x) is an odd function and hence,
`I=underset(-1)overset(1)int g(x)h(x) dx=0`