Find the number of roots of the equation `log_(e)(1+x)-(tan^(-1)x)/(1+x)=0`

Explanation

consider `f(x)=(1+x)log_(e)(1+x)tan^(-1),xgt-1`
`therefore f(x) =log_(e)(1+x)+1-(1)/(1+x^(2))`
`rArr f(x)=(1)/(1+x)+(2x)/(1+x^(2))^(2)`
`=(1)/(1+x)+(2x)/(1+x^(2))^(2)`
`rArr` f(x) =(1)
Sop f(x) is increasing
`therefore` For `xlt0 f(x)ltf(0)`
`rArr f(x)lt0`
`therefore` f(x)is decreasing function
`rArr f(x)gtf(0)`
`therefore f(x)gt0`
`rarr(1+x)log_(e)(1+x)-tan^(-1)xgt0`
`rarrf(x)gt0`
`rArr` f(x) is increasing
`rarr f(x)gtf(0)`
`rArr f(x)gt0`
`log_(e)(1+x)gt(tan^(-1))/(x+1)`
Hence f(x) is continuosu and increasing ,it crossses x axis only once at x=0
Therefore ,equation has only one real root x=0.