What is the integral of (sqrt(4x-12))/2 ?

The integral of (sqrt(4x-12))/2 is 2/3 (x-3)^{3/2}+C

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integral of (sqrt(4x-12))/2

\int \frac{4 x - 12}{2} d x

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

Solution steps

\int \frac{4 x - 12}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int 4 x - 12 d x

Apply u-substitution

: \int \frac{u}{4} d u

= \frac{1}{2} \cdot \int \frac{u}{4} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 x - 12

= \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{3} (4 x - 12)^{\frac{3}{2}}

Simplify

\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{3} (4 x - 12)^{\frac{3}{2}} : \frac{2}{3} (x - 3)^{\frac{3}{2}}

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

Add a constant to the solution

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

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What is the integral of (sqrt(4x-12))/2 ?
The integral of (sqrt(4x-12))/2 is 2/3 (x-3)^{3/2}+C