Evaluate: `int([sqrt(1+x^2)+x]^n)/(sqrt(1+x^2))dx`

Options

A. `(1)/(2(1+x^(2))),x inR`
B. `(1)/(2(1+x^(2))),xgt0`
C. `(1)/(2(1+x^(2))),xlt0`
D. `(-1)/(2(1+x^(2))),xlt0`

Answer

B, D

Explanation

Put `x= tan theta`
`therefore" "tan^(-1)x in(-(pi)/(2),(pi)/(2))`
`therefore" "y=cos^(-1)sqrt((1+|sec theta|)/(2|sec theta|))`
`=cos^(-1)sqrt((cos theta+1)/(2))`
`=cos^(-1)(cos theta//2)`
`={{:(-theta//2",",-pi//2lt theta le0),(theta//2",",0lt theta ltpi//2):}`
`={{:(-(1)/(2)tan^(-1)x",",xle0),((1)/(2)tan^(-1)x",",xgt0):}`
`(dy)/(dx)={{:(-(1)/(2(1+x^(2)))",",xle0),((1)/(2(1+x^(2)))",",xgt0):}`