Evaluate the Integral integral of (x^3-8)^7*3x^2 with respect to x

$\int {(x^{3} - 8)}^{7} \cdot 3 x^{2} d x$

Move $3$ to the left of ${(x^{3} - 8)}^{7}$ .

$\int 3 {(x^{3} - 8)}^{7} x^{2} d x$

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

$3 \int {(x^{3} - 8)}^{7} x^{2} d x$

Let $u = x^{3} - 8$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

Let $u = x^{3} - 8$ . Find $\frac{d u}{d x}$ .

Differentiate $x^{3} - 8$ .

$\frac{d}{d x} [x^{3} - 8]$

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$

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

$3 x^{2} + \frac{d}{d x} [- 8]$

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

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

$3 \int u^{7} \frac{1}{3} d u$

Combine $u^{7}$ and $\frac{1}{3}$ .

$3 \int \frac{u^{7}}{3} d u$

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$3 (\frac{1}{3} \int u^{7} d u)$

Simplify.

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

$\frac{3}{3} \int u^{7} d u$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{3} \int u^{7} d u$

Rewrite the expression.

$1 \int u^{7} d u$

Multiply $\int u^{7} d u$ by $1$ .

$\int u^{7} d u$

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{1}{8} u^{8} + C$

Replace all occurrences of $u$ with $x^{3} - 8$ .

$\frac{1}{8} {(x^{3} - 8)}^{8} + C$