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

The integral of (x-sqrt(x))/(x+sqrt(x)) is

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

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

2 (\frac{2 x ^{\frac{3}{2}} - 3 x}{6 ( x + 1 )} + \frac{x ^{2} - 5 x ^{\frac{3}{2}} + 12 x ln ∣ x + 1 ∣ - 18 x + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}}) + C

Solution steps

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

Apply u-substitution

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

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

Apply Integration By Parts

: \frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 )} - \int - \frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 ) ^{2}} d u

= 2 (\frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 )} - \int - \frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 ) ^{2}} d u)

\int - \frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 ) ^{2}} d u = - \frac{u ^{4} - 5 u ^{3} + 12 u ^{2} ln ∣ u + 1 ∣ - 18 u ^{2} + 24 u ln ∣ u + 1 ∣ - 12 u + 12 ln ∣ u + 1 ∣}{6 ( u + 1 ) ^{2}}

= 2 (\frac{2 u ^{3} - 3 u ^{2}}{6 ( u + 1 )} - (- \frac{u ^{4} - 5 u ^{3} + 12 u ^{2} ln ∣ u + 1 ∣ - 18 u ^{2} + 24 u ln ∣ u + 1 ∣ - 12 u + 12 ln ∣ u + 1 ∣}{6 ( u + 1 ) ^{2}}))

Substitute back

u = x

= 2 (\frac{2 ( x ) ^{3} - 3 ( x ) ^{2}}{6 ( x + 1 )} - (- \frac{( x ) ^{4} - 5 ( x ) ^{3} + 12 ( x ) ^{2} ln ∣ x + 1 ∣ - 18 ( x ) ^{2} + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}}))

Simplify

2 (\frac{2 ( x ) ^{3} - 3 ( x ) ^{2}}{6 ( x + 1 )} - (- \frac{( x ) ^{4} - 5 ( x ) ^{3} + 12 ( x ) ^{2} ln ∣ x + 1 ∣ - 18 ( x ) ^{2} + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}})) : 2 (\frac{2 x ^{\frac{3}{2}} - 3 x}{6 ( x + 1 )} + \frac{x ^{2} - 5 x ^{\frac{3}{2}} + 12 x ln ∣ x + 1 ∣ - 18 x + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}})

= 2 (\frac{2 x ^{\frac{3}{2}} - 3 x}{6 ( x + 1 )} + \frac{x ^{2} - 5 x ^{\frac{3}{2}} + 12 x ln ∣ x + 1 ∣ - 18 x + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}})

Add a constant to the solution

= 2 (\frac{2 x ^{\frac{3}{2}} - 3 x}{6 ( x + 1 )} + \frac{x ^{2} - 5 x ^{\frac{3}{2}} + 12 x ln ∣ x + 1 ∣ - 18 x + 24 x ln ∣ x + 1 ∣ - 12 x + 12 ln ∣ x + 1 ∣}{6 ( x + 1 ) ^{2}}) + C

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