What is the integral of 1/(1+sqrt(1+2x)) ?

The integral of 1/(1+sqrt(1+2x)) is 1+sqrt(1+2x)-ln|1+sqrt(1+2x)|+C

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integral of 1/(1+sqrt(1+2x))

\int \frac{1}{1 + 1 + 2 x} d x

1 + 1 + 2 x - ln 1 + 1 + 2 x + C

Solution steps

\int \frac{1}{1 + 1 + 2 x} d x

Apply u-substitution

= \int \frac{u}{1 + u} d u

Apply u-substitution

= \int \frac{v - 1}{v} d v

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \int 1 - \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d v - \int \frac{1}{v} d v

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= v - ln ∣ v ∣

Substitute back

= 1 + 1 + 2 x - ln 1 + 1 + 2 x

Add a constant to the solution

= 1 + 1 + 2 x - ln 1 + 1 + 2 x + C

What is the integral of 1/(1+sqrt(1+2x)) ?
The integral of 1/(1+sqrt(1+2x)) is 1+sqrt(1+2x)-ln|1+sqrt(1+2x)|+C