Calculus Examples

$\int \frac{12 x^{2}}{4 x^{3} - 6} d x$

Step 1

Cancel the common factor of $12$ and $4 x^{3} - 6$ .

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Step 1.1

Factor $2$ out of $12 x^{2}$ .

$\int \frac{2 (6 x^{2})}{4 x^{3} - 6} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $4 x^{3}$ .

$\int \frac{2 (6 x^{2})}{2 (2 x^{3}) - 6} d x$

Step 1.2.2

Factor $2$ out of $- 6$ .

$\int \frac{2 (6 x^{2})}{2 (2 x^{3}) + 2 (- 3)} d x$

Step 1.2.3

Factor $2$ out of $2 (2 x^{3}) + 2 (- 3)$ .

$\int \frac{2 (6 x^{2})}{2 (2 x^{3} - 3)} d x$

Step 1.2.4

Cancel the common factor.

$\int \frac{2 (6 x^{2})}{2 (2 x^{3} - 3)} d x$

Step 1.2.5

Rewrite the expression.

$\int \frac{6 x^{2}}{2 x^{3} - 3} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [- 3]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$6 x^{2} + 0$

Step 2.1.4.2

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

$6 x^{2}$

Step 2.2

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

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

Step 3

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

$\ln (| u |) + C$

Step 4

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

$\ln (| 2 x^{3} - 3 |) + C$