Calculus Examples

$\int \frac{2 x - 4}{3 x^{2} - 12 x + 1} d x$

Step 1

Factor $2$ out of $2 x - 4$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$\int \frac{2 (x) - 4}{3 x^{2} - 12 x + 1} d x$

Step 1.2

Factor $2$ out of $- 4$ .

$\int \frac{2 x + 2 \cdot - 2}{3 x^{2} - 12 x + 1} d x$

Step 1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\int \frac{2 (x - 2)}{3 x^{2} - 12 x + 1} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{x - 2}{3 x^{2} - 12 x + 1} d x$

Step 3

Let $u = 3 x^{2} - 12 x + 1$ . Then $d u = (6 x - 12) d x$ , so $\frac{1}{6} d u = (x - 2) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 3 x^{2} - 12 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $3 x^{2} - 12 x + 1$ .

$\frac{d}{d x} [3 x^{2} - 12 x + 1]$

Step 3.1.2

By the Sum Rule, the derivative of $3 x^{2} - 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$

Step 3.1.3

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

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Step 3.1.3.1

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

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$

Step 3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$

Step 3.1.3.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$

Step 3.1.4

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Step 3.1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$6 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.1.4.2

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

$6 x - 12 \cdot 1 + \frac{d}{d x} [1]$

Step 3.1.4.3

Multiply $- 12$ by $1$ .

$6 x - 12 + \frac{d}{d x} [1]$

Step 3.1.5

Differentiate using the Constant Rule.

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Step 3.1.5.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$6 x - 12 + 0$

Step 3.1.5.2

Add $6 x - 12$ and $0$ .

$6 x - 12$

Step 3.2

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

$2 \int \frac{1}{u} \cdot \frac{1}{6} d u$

Step 4

Simplify.

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Step 4.1

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

$2 \int \frac{1}{u \cdot 6} d u$

Step 4.2

Move $6$ to the left of $u$ .

$2 \int \frac{1}{6 u} d u$

Step 5

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

$2 (\frac{1}{6} \int \frac{1}{u} d u)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{6}$ and $2$ .

$\frac{2}{6} \int \frac{1}{u} d u$

Step 6.2

Cancel the common factor of $2$ and $6$ .

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Step 6.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} \int \frac{1}{u} d u$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} \int \frac{1}{u} d u$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} \int \frac{1}{u} d u$

Step 6.2.2.3

Rewrite the expression.

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

Step 7

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

$\frac{1}{3} (\ln (| u |) + C)$

Step 8

Simplify.

$\frac{1}{3} \ln (| u |) + C$

Step 9

Replace all occurrences of $u$ with $3 x^{2} - 12 x + 1$ .

$\frac{1}{3} \ln (| 3 x^{2} - 12 x + 1 |) + C$