Evaluate `int(e^x(dx))/(sqrt(1+e^(x)+e^(2x)))`

Answer

`-log_(e)|(e^(-x)+(1)/(2))+sqrt(e^(-2x)+e^(-x)+1)|+c`

Explanation

`I= int(dx)/(sqrt(1+e^(x)+e^(2x)))=int(e^(x)dx)/(e^(x)sqrt(1+e^(x)+e^(2x)))`
`=int(dt)/(t sqrt(1+t+t^(2)))`
` " Put " t=(1)/(y) `
` :. I=int(-(dy)/(y^(2)))/((1)/(y)sqrt(1+(1)/(y)+(1)/(y^(2))))`
`=-int(dy)/(sqrt(y^(2)+y+1))`
`=-int(dy)/(sqrt((y+(1)/(2))^(2)+(3)/(4))) `
`=-log_(e)|(y+(1)/(2))+sqrt((y+(1)/(2))^(2)+(3)/(4))|+c`
`=-log_(e)|(e^(-x)+(1)/(2))+sqrt(e^(-2x)+e^(-x)+1)|+c`