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

The integral of (sqrt(x))/(1+xsqrt(x)) is 2/3 ln|1+xsqrt(x)|+C

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

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

\frac{2}{3} ln 1 + x x + C

Solution steps

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

Apply u-substitution

: \int \frac{2}{3 u} d u

= \int \frac{2}{3 u} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{2}{3} ln ∣ u ∣

Substitute back

u = 1 + x x

= \frac{2}{3} ln 1 + x x

Add a constant to the solution

= \frac{2}{3} ln 1 + x x + C

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What is the integral of (sqrt(x))/(1+xsqrt(x)) ?
The integral of (sqrt(x))/(1+xsqrt(x)) is 2/3 ln|1+xsqrt(x)|+C