What is the integral of (3x)/(sqrt(2x+4)) ?

The integral of (3x)/(sqrt(2x+4)) is (x-4)sqrt(2x+4)+C

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integral of (3x)/(sqrt(2x+4))

\int \frac{3 x}{2 x + 4} d x

(x - 4) 2 x + 4 + C

Solution steps

\int \frac{3 x}{2 x + 4} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{2 x + 4} d x

Apply u-substitution

= 3 \cdot \int \frac{u ^{2} - 4}{2} d u

Take the constant out:

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

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

= 3 \cdot \frac{1}{2} (\int u^{2} d u - \int 4 d u)

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

\int 4 d u = 4 u

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

Substitute back

u = 2 x + 4

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

Simplify

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

= (x - 4) 2 x + 4

Add a constant to the solution

= (x - 4) 2 x + 4 + C

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