Calculus Examples

$\int \frac{6 x^{2} - 16 x + 2}{x^{3} - 4 x^{2} + x + 6} d x$

Step 1

Factor $2$ out of $6 x^{2} - 16 x + 2$ .

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Step 1.1

Factor $2$ out of $6 x^{2}$ .

$\int \frac{2 (3 x^{2}) - 16 x + 2}{x^{3} - 4 x^{2} + x + 6} d x$

Step 1.2

Factor $2$ out of $- 16 x$ .

$\int \frac{2 (3 x^{2}) + 2 (- 8 x) + 2}{x^{3} - 4 x^{2} + x + 6} d x$

Step 1.3

Factor $2$ out of $2$ .

$\int \frac{2 (3 x^{2}) + 2 (- 8 x) + 2 (1)}{x^{3} - 4 x^{2} + x + 6} d x$

Step 1.4

Factor $2$ out of $2 (3 x^{2}) + 2 (- 8 x)$ .

$\int \frac{2 (3 x^{2} - 8 x) + 2 (1)}{x^{3} - 4 x^{2} + x + 6} d x$

Step 1.5

Factor $2$ out of $2 (3 x^{2} - 8 x) + 2 (1)$ .

$\int \frac{2 (3 x^{2} - 8 x + 1)}{x^{3} - 4 x^{2} + x + 6} d x$

Step 2

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

$2 \int \frac{3 x^{2} - 8 x + 1}{x^{3} - 4 x^{2} + x + 6} d x$

Step 3

Let $u = x^{3} - 4 x^{2} + x + 6$ . Then $d u = (3 x^{2} - 8 x + 1) d x$ , so $\frac{1}{3 x^{2} - 8 x + 1} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x^{3} - 4 x^{2} + x + 6$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x^{3} - 4 x^{2} + x + 6$ .

$\frac{d}{d x} [x^{3} - 4 x^{2} + x + 6]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $x^{3} - 4 x^{2} + x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 3.1.2.2

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

$3 x^{2} + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 3.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 3.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [6]$

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 $n = 2$ .

$3 x^{2} - 4 (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 3.1.3.3

Multiply $2$ by $- 4$ .

$3 x^{2} - 8 x + \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 3.1.4

Differentiate.

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Step 3.1.4.1

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

$3 x^{2} - 8 x + 1 + \frac{d}{d x} [6]$

Step 3.1.4.2

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

$3 x^{2} - 8 x + 1 + 0$

Step 3.1.4.3

Add $3 x^{2} - 8 x + 1$ and $0$ .

$3 x^{2} - 8 x + 1$

Step 3.2

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

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

Step 4

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

$2 (\ln (| u |) + C)$

Step 5

Simplify.

$2 \ln (| u |) + C$

Step 6

Replace all occurrences of $u$ with $x^{3} - 4 x^{2} + x + 6$ .

$2 \ln (| x^{3} - 4 x^{2} + x + 6 |) + C$