What is the integral of x/(sqrt(x-12)) ?

The integral of x/(sqrt(x-12)) is 2(1/3 (x-12)^{3/2}+12sqrt(x-12))+C

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integral of x/(sqrt(x-12))

\int \frac{x}{x - 12} d x

2 (\frac{1}{3} (x - 12)^{\frac{3}{2}} + 12 x - 12) + C

Solution steps

\int \frac{x}{x - 12} d x

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

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

\int 12 d u = 12 u

= 2 (\frac{u ^{3}}{3} + 12 u)

Substitute back

u = x - 12

= 2 (\frac{( x - 12 ) ^{3}}{3} + 12 x - 12)

Simplify

2 (\frac{( x - 12 ) ^{3}}{3} + 12 x - 12) : 2 (\frac{1}{3} (x - 12)^{\frac{3}{2}} + 12 x - 12)

= 2 (\frac{1}{3} (x - 12)^{\frac{3}{2}} + 12 x - 12)

Add a constant to the solution

= 2 (\frac{1}{3} (x - 12)^{\frac{3}{2}} + 12 x - 12) + C

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What is the integral of x/(sqrt(x-12)) ?
The integral of x/(sqrt(x-12)) is 2(1/3 (x-12)^{3/2}+12sqrt(x-12))+C