Find the range of f(x)=`(1)/(pi)sin^(-1)x+tan^(-1)+(x+1)/(x^(2)+2x+5)`

Explanation

Clearly domain of the functions is [-1.1] Both `sin^(-1)x` and `tan^(-1)x` are increasing functions.
Also `(d)/(dx)(x+1)/(x^(2)+2x+5)=(x^(2)+2x+5-(2x+2)(x+1))/(x^(2)+2x+5)^(2)`
`=-(x^(2)(-2x+3))/(x^(2)+2x+5)^(2)`
`=-((x-1)(x+3))/(x^(2)+2x+5)^(2)geforall x in [-1,1]`
`therefore` f(x) is an increasing function
Therefore range of f(x)is [f(-1),f(1)]
`f(-1)=(1)/(pi)(pi)/(2)-(pi)/(4)+0=(-3)/(4)`
`f(1)=(1)/(pi)(pi)/(2)+(pi)/(4)+9/4=1`
Hence range of f(x) is `[(3)/(4),1]`