Evaluate the Integral integral of 2(2x-6)^8 with respect to x

$\int 2 {(2 x - 6)}^{8} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int {(2 x - 6)}^{8} d x$

Step 2

Let $u = 2 x - 6$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Let $u = 2 x - 6$ . Find $\frac{d u}{d x}$ .

Differentiate $2 x - 6$ .

$\frac{d}{d x} [2 x - 6]$

By the Sum Rule, the derivative of $2 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$

Evaluate $\frac{d}{d x} [2 x]$ .

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 6]$

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 6]$

Differentiate using the Constant Rule.

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$2 + 0$

Add $2$ and $0$ .

$2$

Rewrite the problem using $u$ and $d u$ .

$2 \int u^{8} \frac{1}{2} d u$

Step 3

Combine $u^{8}$ and $\frac{1}{2}$ .

$2 \int \frac{u^{8}}{2} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int u^{8} d u)$

Step 5

Simplify.

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

$\frac{2}{2} \int u^{8} d u$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{2} \int u^{8} d u$

Rewrite the expression.

$1 \int u^{8} d u$

Multiply $\int u^{8} d u$ by $1$ .

$\int u^{8} d u$

Step 6

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$\frac{1}{9} u^{9} + C$

Step 7

Replace all occurrences of $u$ with $2 x - 6$ .

$\frac{1}{9} {(2 x - 6)}^{9} + C$

Evaluate the Integral integral of 2(2x-6)^8 with respect to x

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