What is the integral of (3+sqrt(x))/(3-sqrt(x)) ?

The integral of (3+sqrt(x))/(3-sqrt(x)) is -x-12sqrt(x)-36ln|-sqrt(x)+3|+C

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integral of (3+sqrt(x))/(3-sqrt(x))

\int \frac{3 + x}{3 - x} d x

- x - 12 x - 36 ln - x + 3 + C

Solution steps

\int \frac{3 + x}{3 - x} d x

Apply u-substitution

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

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

Long division

\frac{u ( 3 + u )}{3 - u} : - u - 6 + \frac{18}{- u + 3}

= 2 \cdot \int - u - 6 + \frac{18}{- u + 3} 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 6 d u + \int \frac{18}{- u + 3} d u)

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

\int 6 d u = 6 u

\int \frac{18}{- u + 3} d u = - 18 ln ∣ - u + 3 ∣

= 2 (- \frac{u ^{2}}{2} - 6 u - 18 ln ∣ - u + 3 ∣)

Substitute back

u = x

= 2 (- \frac{( x ) ^{2}}{2} - 6 x - 18 ln - x + 3)

Simplify

2 (- \frac{( x ) ^{2}}{2} - 6 x - 18 ln - x + 3) : - x - 12 x - 36 ln - x + 3

= - x - 12 x - 36 ln - x + 3

Add a constant to the solution

= - x - 12 x - 36 ln - x + 3 + C

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What is the integral of (3+sqrt(x))/(3-sqrt(x)) ?
The integral of (3+sqrt(x))/(3-sqrt(x)) is -x-12sqrt(x)-36ln|-sqrt(x)+3|+C