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प्रश्न
Integrate the following with respect to x:
`sin^-1 ((2x)/(1 + x^2))`
उत्तर
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"`
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