Question

Integrate e^(sqrt x )

Answer 1

We will use partial integration:
`\int {u} \, dv= u * v - \int {v} \, du` First we will substitute:
`u=e^( √(x) ) , dv = dx, du = e^( √(x) ) * (1)/(2 √(x) ) dx, v = x`...
`=x * e^( √(x) ) - \int {xe^( √(x) ) (1)/(2 √(x) ) } \, dx`
Another substitution:
`t= √(x) . t^(2) =x, dt= (dx)/(2 √(x) )`

`\int {t^(2)e^(2) } \, dt =t^(2) e^( √(x) ) - \int {te^( √(x) ) } \, dt =t^(2) e ^(t) -2te^(t) +2 e^(t)`
Finally: ...=
`xe^( √(x) ) -xe^( √(x) )-2 √(x) e^( √(x) ) -2e^( √(x) ) = 2( √(x) -1)*e^( √(x) ) +C`
Thank you.