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

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

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

\int \frac{x}{x ( 1 + x )} d x

2 ln 1 + x + C

Solution steps

\int \frac{x}{x ( 1 + x )} d x

Simplify

\frac{x}{x ( 1 + x )} : \frac{1}{x ( 1 + x )}

= \int \frac{1}{x ( 1 + x )} d x

Apply u-substitution

: \int \frac{2}{u} d u

= \int \frac{2}{u} d u

Take the constant out:

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

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 1 + x

= 2 ln 1 + x

Add a constant to the solution

= 2 ln 1 + x + C

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