# 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

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