Answer 1

Let I = `int "e"^"x"/sqrt("e"^"2x" + 4"e"^"x" + 13)` dx

`= int "e"^"x"/sqrt(("e"^"x")^2 +  4"e"^"x"  + 13)` dx

Put e^x = t

∴ e^x  dx = dt

∴ I = `"dt"/(sqrt("t"^2 + 4"t" + 13))`

`= int 1/sqrt("t"^2 + 4"t" + 4 - 4 + 13)` dt

`= int 1/(sqrt(("t + 2")^2 + 9))` dt

`= int 1/(sqrt(("t" + 2)^2 + (3)^2))` dt

`= log |"t" + 2 + sqrt(("t" + 2)^2 + (3)^2)|` + c

`= log |"t" + 2 + sqrt("t"^2 + 4"t" + 13)|`

∴ I = `log |"e"^"x" + 2 + sqrt("e"^"2x" + 4"e"^"x" + 13)|`