What is the integral of (x+2)sqrt((3x^2+12x)) ?

The integral of (x+2)sqrt((3x^2+12x)) is sqrt(1/3 (x^2+4x)^3)+C

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integral of (x+2)sqrt((3x^2+12x))

\int (x + 2) (3 x^{2} + 12 x) d x

\frac{1}{3} (x^{2} + 4 x)^{3} + C

Solution steps

\int (x + 2) 3 x^{2} + 12 x d x

(x + 2) 3 x^{2} + 12 x = 3 (x + 2) x^{2} + 4 x

= \int 3 (x + 2) x^{2} + 4 x d x

Take the constant out:

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

= 3 \cdot \int (x + 2) x^{2} + 4 x d x

Apply u-substitution

: \int u^{2} d u

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

Apply the Power Rule

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

= 3 \frac{u ^{3}}{3}

Substitute back

u = x^{2} + 4 x

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (x y^{2} + y e^{x^{2}} + 5)

y = ln (x + 5)

x \to 2 lim (x^{2} + 6)

\frac{( 4 x ^{2} + 4 x + 6 )}{x}

\int \frac{x + 5}{2 x + 3} d x

What is the integral of (x+2)sqrt((3x^2+12x)) ?
The integral of (x+2)sqrt((3x^2+12x)) is sqrt(1/3 (x^2+4x)^3)+C