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

The integral of x/(1+sqrt(1+x)) is 2/3 (1+x)^{3/2}-1-x+C

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

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

\frac{2}{3} (1 + x)^{\frac{3}{2}} - 1 - x + C

Solution steps

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

Apply u-substitution

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

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

= \int u - 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u - \int 1 d u

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

\int 1 d u = u

= \frac{2}{3} u^{\frac{3}{2}} - u

Substitute back

u = 1 + x

= \frac{2}{3} (1 + x)^{\frac{3}{2}} - (1 + x)

Simplify

= \frac{2}{3} (1 + x)^{\frac{3}{2}} - 1 - x

Add a constant to the solution

= \frac{2}{3} (1 + x)^{\frac{3}{2}} - 1 - x + C

What is the integral of x/(1+sqrt(1+x)) ?
The integral of x/(1+sqrt(1+x)) is 2/3 (1+x)^{3/2}-1-x+C