Evaluate `int tan^(-1)(secx+tanx)dx,-pi//2ltxlt pi//2`

Answer

`(pi)/(4)x+(x^(2))/(4)+C`

Explanation

`int tan^(-1)(secx+tanx)dx`
`=int "tan"^(-1)((1+sinx)/(cosx))dx`
`=int "tan"^(-1){(1-"cos"((pi)/(2)+x))/("sin"((pi)/(2)+x))}dx`
`=int "tan"^(-1){(2"sin"^(2)((pi)/(4)+(x)/(2)))/(2"sin"((pi)/(4)+(x)/(2))"cos"((pi)/(4)+(x)/(2)))}dx`
`=int tan^(-1){tan((pi)/(4)+(x)/(2))}dx`
`=int (pi)/(4)+(x)/(2)dx=(pi)/(4)int1*dx+(1)/(2)int xdx=(pi)/(4)x+(x^(2))/(4)+C`