Question

Integrate the following with respect to x:

`sin^-1 ((2x)/(1 + x^2))`

Answer 1

I = `int sin^-1 ((2x)/(1 + x^2))  ""d"x`

Put x = tan θ

dx  =  sec²θ dθ

I = `int sin^-1 ((2tantheta)/(1 + tan^2theta)) sec^2theta  "d"theta`

= `int sin^-1 (sin2theta) sec^2theta  "d"theta`

= `int 2theta sec^2theta  "d"theta`

= `2int (theta) (sec^2theta  "d"theta)`  ........[Rater Example 11.34]

I = `2[thetatantheta - int tan theta  "d"theta]`

= `2theta tan theta - 2 log |sectheta| + "c"`  ......`(sec theta = sqrt(1 + tan^2theta))`

`int sin^-1 ((2x)/(1 + x^2))  "d"x = 2x tan^-1x - 2log |sqrt(1 + x^2)| + "c"`

= `2[x tan^-1x - log |sqrt(1 + x^2)|] + "c"`