Evaluate the Integral integral of x^2(2x^3+2)^8 with respect to x

$\int x^{2} {(2 x^{3} + 2)}^{8} d x$

Step 1

Let $u = 2 x^{3} + 2$ . Then $d u = 6 x^{2} d x$ , so $\frac{1}{6} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

Let $u = 2 x^{3} + 2$ . Find $\frac{d u}{d x}$ .

Differentiate $2 x^{3} + 2$ .

$\frac{d}{d x} [2 x^{3} + 2]$

By the Sum Rule, the derivative of $2 x^{3} + 2$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [2]$

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$2 (3 x^{2}) + \frac{d}{d x} [2]$

Multiply $3$ by $2$ .

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

Differentiate using the Constant Rule.

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$6 x^{2} + 0$

Add $6 x^{2}$ and $0$ .

$6 x^{2}$

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

$\int u^{8} \frac{1}{6} d u$

Step 2

Combine $u^{8}$ and $\frac{1}{6}$ .

$\int \frac{u^{8}}{6} d u$

Step 3

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int u^{8} d u$

Step 4

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

$\frac{1}{6} (\frac{1}{9} u^{9} + C)$

Step 5

Simplify.

Rewrite $\frac{1}{6} (\frac{1}{9} u^{9} + C)$ as $\frac{1}{6} \cdot \frac{1}{9} u^{9} + C$ .

$\frac{1}{6} \cdot \frac{1}{9} u^{9} + C$

Simplify.

Multiply $\frac{1}{6}$ by $\frac{1}{9}$ .

$\frac{1}{6 \cdot 9} u^{9} + C$

Multiply $6$ by $9$ .

$\frac{1}{54} u^{9} + C$

Step 6

Replace all occurrences of $u$ with $2 x^{3} + 2$ .

$\frac{1}{54} {(2 x^{3} + 2)}^{9} + C$