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

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

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

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

\frac{x}{1 + x} (1 + x) - ln 1 + x + x + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 1}{u} d u

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

Apply Integration By Parts

: \frac{u - 1}{u} u - \int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u

= \frac{u - 1}{u} u - \int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u

\int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u = ln u + u - 1

= \frac{u - 1}{u} u - ln u + u - 1

Substitute back

u = 1 + x

= \frac{1 + x - 1}{1 + x} (1 + x) - ln 1 + x + 1 + x - 1

Simplify

= \frac{x}{1 + x} (1 + x) - ln 1 + x + x

Add a constant to the solution

= \frac{x}{1 + x} (1 + x) - ln 1 + x + x + C

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