`int(1)/(sqrt(1+4x^(2))`dx

Explanation

`int(1)/(sqrt(1+4x^(2)))dx`
`=(1)/(2)int(1)/(sqrt(x^(2)+((1)/(2))^(2)))dx`
`=(1)/(2)log|x+sqrt(x^(2)+((1)/(2))^(2))|+c_(1)`
`=(1)/(2)log|(2x+sqrt(4x^(2)+1)/(2)|+c_(1)`
`=(1)/(2)log|2x+sqrt(4x^(2)+1)|-(1)/(2)log2+c_(1)`
`=(1)/(2)log|2x+sqrt(4x^(2)+1)|+c`