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

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

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integral of 1/(sqrt(x)(2+\sqrt{x))}

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

2 ln 2 + x + C

Solution steps

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

Apply u-substitution

= \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 = 2 + x

= 2 ln 2 + x

Add a constant to the solution

= 2 ln 2 + x + C

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