`int 1/(1 + sqrt(2x)) dx` Find the indefinite integral by u substitution. (let u be the denominator of the integral)

Solving for indefinite integral using u-substitution follows:

`int f(g(x))*g'(x) dx = int f(u) du` where we let` u = g(x)` .

In this case, it is stated that to let u be the denominator of integral which means let:

`u = 1+sqrt(2x).`

This can be rearrange into `sqrt(2x) = u -1`

Finding the derivative of u :  `du = 1/sqrt(2x) dx`

Substituting `sqrt(2x)= u-1` into `du = 1/sqrt(2x)dx` becomes:

`du = 1/(u-1)dx`

Rearranged into `(u-1) du =dx`

Applying u-substitution using` u = 1+sqrt(2x) `  and `(u-1)du = du` :

`int 1/(1+sqrt(2x)) dx = int (u-1)/u *du`

Express into two separate fractions:

`int (u-1)/u *du = int ( u/u -1/u)du`

                      ` = int (1 - 1/u)du`

Applying `int (f(x) -g(x))dx = int f(x) dx - int g(x) dx` :

`int (1 - 1/u)du = int 1 du - int 1/udu`

                       `= u - ln|u| +C`

Substitute `u = 1+sqrt(2x) `  to the `u - ln|u| +C` :

`u - ln|u| +C =1+sqrt(2x) -ln|1+sqrt(2x) |+C`