Evaluate the Integral integral of (3x+6)(x+2) with respect to x

$\int (3 x + 6) (x + 2) d x$

Step 1

Simplify.

Apply the distributive property.

$\int 3 x (x + 2) + 6 (x + 2) d x$

Apply the distributive property.

$\int 3 x \cdot x + 3 x \cdot 2 + 6 (x + 2) d x$

Apply the distributive property.

$\int 3 x \cdot x + 3 x \cdot 2 + 6 x + 6 \cdot 2 d x$

Move $x$ .

$\int 3 x \cdot x + 3 \cdot 2 x + 6 x + 6 \cdot 2 d x$

Raise $x$ to the power of $1$ .

$\int 3 (x^{1} x) + 3 \cdot 2 x + 6 x + 6 \cdot 2 d x$

Raise $x$ to the power of $1$ .

$\int 3 (x^{1} x^{1}) + 3 \cdot 2 x + 6 x + 6 \cdot 2 d x$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 3 x^{1 + 1} + 3 \cdot 2 x + 6 x + 6 \cdot 2 d x$

Add $1$ and $1$ .

$\int 3 x^{2} + 3 \cdot 2 x + 6 x + 6 \cdot 2 d x$

Multiply $3$ by $2$ .

$\int 3 x^{2} + 6 x + 6 x + 6 \cdot 2 d x$

Multiply $6$ by $2$ .

$\int 3 x^{2} + 6 x + 6 x + 12 d x$

Add $6 x$ and $6 x$ .

$\int 3 x^{2} + 12 x + 12 d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 x^{2} d x + \int 12 x d x + \int 12 d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{2} d x + \int 12 x d x + \int 12 d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$3 (\frac{1}{3} x^{3} + C) + \int 12 x d x + \int 12 d x$

Step 5

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$3 (\frac{1}{3} x^{3} + C) + 12 \int x d x + \int 12 d x$

Step 6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Simplify.

$x^{3} + 6 x^{2} + 12 x + C$

Evaluate the Integral integral of (3x+6)(x+2) with respect to x

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