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

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

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

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

x - 2 x + 2 ln x + 1 - 3 + C

Solution steps

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

Simplify

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

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

Apply u-substitution

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

Expand

\frac{u ^{2} - 2 u + 1}{u} : u - 2 + \frac{1}{u}

= 2 \cdot \int u - 2 + \frac{1}{u} 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

= 2 (\int u d u - \int 2 d u + \int \frac{1}{u} d u)

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} - 2 u + ln ∣ u ∣)

Substitute back

u = x + 1

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

Simplify

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

= x - 2 x + 2 ln x + 1 - 3

Add a constant to the solution

= x - 2 x + 2 ln x + 1 - 3 + C

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