What is the integral of (6-6x)/(1-sqrt(x)) ?

The integral of (6-6x)/(1-sqrt(x)) is 12(1/3 x^{3/2}+x/2)+C

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integral of (6-6x)/(1-sqrt(x))

\int \frac{6 - 6 x}{1 - x} d x

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

Solution steps

\int \frac{6 - 6 x}{1 - x} d x

Apply u-substitution

= \int 12 u (u + 1) d u

Take the constant out:

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

= 12 \cdot \int u (u + 1) d u

Expand

u (u + 1) : u^{2} + u

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

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

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

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

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

Substitute back

u = x

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of (6-6x)/(1-sqrt(x)) ?
The integral of (6-6x)/(1-sqrt(x)) is 12(1/3 x^{3/2}+x/2)+C