Calculus Examples

$\int (\frac{7 x^{6}}{x^{7} + 2}) d x$

Step 1

Remove parentheses.

$\int \frac{7 x^{6}}{x^{7} + 2} d x$

Step 2

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int \frac{x^{6}}{x^{7} + 2} d x$

Step 3

Let $u = x^{7} + 2$ . Then $d u = 7 x^{6} d x$ , so $\frac{1}{7} d u = x^{6} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x^{7} + 2$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x^{7} + 2$ .

$\frac{d}{d x} [x^{7} + 2]$

Step 3.1.2

By the Sum Rule, the derivative of $x^{7} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{7}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{7}] + \frac{d}{d x} [2]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + \frac{d}{d x} [2]$

Step 3.1.4

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

$7 x^{6} + 0$

Step 3.1.5

Add $7 x^{6}$ and $0$ .

$7 x^{6}$

Step 3.2

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

$7 \int \frac{1}{u} \cdot \frac{1}{7} d u$

Step 4

Simplify.

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Step 4.1

Multiply $\frac{1}{u}$ by $\frac{1}{7}$ .

$7 \int \frac{1}{u \cdot 7} d u$

Step 4.2

Move $7$ to the left of $u$ .

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

Step 5

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

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

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{7}$ and $7$ .

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

Step 6.2

Cancel the common factor of $7$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

$1 \int \frac{1}{u} d u$

Step 6.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\int \frac{1}{u} d u$

Step 7

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| u |) + C$

Step 8

Replace all occurrences of $u$ with $x^{7} + 2$ .

$\ln (| x^{7} + 2 |) + C$