Calculus Examples

$\int {(x^{8} + 6)}^{7} (8 x^{7}) d x$

Step 1

Move $8$ to the left of ${(x^{8} + 6)}^{7}$ .

$\int 8 {(x^{8} + 6)}^{7} x^{7} d x$

Step 2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int {(x^{8} + 6)}^{7} x^{7} d x$

Step 3

Let $u = x^{8} + 6$ . Then $d u = 8 x^{7} d x$ , so $\frac{1}{8} d u = x^{7} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x^{8} + 6$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x^{8} + 6$ .

$\frac{d}{d x} [x^{8} + 6]$

Step 3.1.2

By the Sum Rule, the derivative of $x^{8} + 6$ with respect to $x$ is $\frac{d}{d x} [x^{8}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x^{8}] + \frac{d}{d x} [6]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$8 x^{7} + \frac{d}{d x} [6]$

Step 3.1.4

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

$8 x^{7} + 0$

Step 3.1.5

Add $8 x^{7}$ and $0$ .

$8 x^{7}$

Step 3.2

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

$8 \int u^{7} \frac{1}{8} d u$

Step 4

Combine $u^{7}$ and $\frac{1}{8}$ .

$8 \int \frac{u^{7}}{8} d u$

Step 5

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$8 (\frac{1}{8} \int u^{7} d u)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{8}$ and $8$ .

$\frac{8}{8} \int u^{7} d u$

Step 6.2

Cancel the common factor of $8$ .

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Step 6.2.1

Cancel the common factor.

$\frac{8}{8} \int u^{7} d u$

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

$\int u^{7} d u$

Step 7

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

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

Step 8

Replace all occurrences of $u$ with $x^{8} + 6$ .

$\frac{1}{8} {(x^{8} + 6)}^{8} + C$