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

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

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integral of 1/(1+sqrt(u))

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

2 (1 + u - ln 1 + u) + C

Solution steps

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

Apply u-substitution

= \int \frac{2 ( v - 1 )}{v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{v - 1}{v} d v

Expand

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

= 2 \cdot \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

= 2 (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 2 (v - ln ∣ v ∣)

Substitute back

v = 1 + u

= 2 (1 + u - ln 1 + u)

Add a constant to the solution

= 2 (1 + u - ln 1 + u) + C

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