What is the integral of (sqrt(x))/(sqrt(x)-6) ?

The integral of (sqrt(x))/(sqrt(x)-6) is x+12sqrt(x)+72ln|sqrt(x)-6|-108+C

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integral of (sqrt(x))/(sqrt(x)-6)

\int \frac{x}{x - 6} d x

x + 12 x + 72 ln x - 6 - 108 + C

Solution steps

\int \frac{x}{x - 6} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} + 12 u + 36 )}{u} d u

= \int \frac{2 ( u ^{2} + 12 u + 36 )}{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} + 12 u + 36}{u} d u

Expand

\frac{u ^{2} + 12 u + 36}{u} : u + 12 + \frac{36}{u}

= 2 \cdot \int u + 12 + \frac{36}{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 12 d u + \int \frac{36}{u} d u)

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

\int 12 d u = 12 u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} + 12 u + 36 ln ∣ u ∣)

Substitute back

u = x - 6

= 2 (\frac{( x - 6 ) ^{2}}{2} + 12 (x - 6) + 36 ln x - 6)

Simplify

2 (\frac{( x - 6 ) ^{2}}{2} + 12 (x - 6) + 36 ln x - 6) : x + 12 x + 72 ln x - 6 - 108

= x + 12 x + 72 ln x - 6 - 108

Add a constant to the solution

= x + 12 x + 72 ln x - 6 - 108 + C

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What is the integral of (sqrt(x))/(sqrt(x)-6) ?
The integral of (sqrt(x))/(sqrt(x)-6) is x+12sqrt(x)+72ln|sqrt(x)-6|-108+C