`int_(-1)^(1)log(x+sqrt(x^(2)+1))dx=?`

Options

A. 0
B. log 2
C. `"log"(1)/(2)`
D. none of these

Answer

A

Explanation

Let `f(x)=log(x+sqrt(1+x^(2)))`
and replacing x by -x, we get
`f(-x)=log(sqrt(1+x^(2))-x)`
`=log(sqrt(1+x^(2))-x)((sqrt(1+x^(2))+x))/((sqrt(1+x^(2))+x))`
`="log"([(1+x^(2))-x^(2)])/((sqrt(1+x^(2))+x))`
`=log1-lo(sqrt(1+x^(2))+x)`
`=-log(sqrt(1+x^(2))+x)impliesf(-x)=-f(x)`
Hence, f(x) is an odd function.
`int_(-1)^(1)log(x+sqrt(1+x^(2)))dx=0`.