Calculus Examples

$\int_{4}^{6} \frac{x^{2} + 2}{x - 2} d x$

Step 1

Divide $x^{2} + 2$ by $x - 2$ .

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Step 1.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{2}$

$0 x$

$2$

Step 1.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$

Step 1.3

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			+	$x^{2}$	-	$2 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - 2 x$

				$x$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$

Step 1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$

Step 1.6

Pull the next terms from the original dividend down into the current dividend.

				$x$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$	+	$2$

Step 1.7

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

				$x$	+	$2$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$	+	$2$

Step 1.8

Multiply the new quotient term by the divisor.

				$x$	+	$2$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$	+	$2$
					+	$2 x$	-	$4$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x - 4$

				$x$	+	$2$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$	+	$2$
					-	$2 x$	+	$4$

Step 1.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$	+	$2$
$x$	-	$2$		$x^{2}$	+	$0 x$	+	$2$
			-	$x^{2}$	+	$2 x$
					+	$2 x$	+	$2$
					-	$2 x$	+	$4$
							+	$6$

Step 1.11

The final answer is the quotient plus the remainder over the divisor.

$\int_{4}^{6} x + 2 + \frac{6}{x - 2} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{4}^{6} x d x + \int_{4}^{6} 2 d x + \int_{4}^{6} \frac{6}{x - 2} d x$

Step 3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2}]_{4}^{6} + \int_{4}^{6} 2 d x + \int_{4}^{6} \frac{6}{x - 2} d x$

Step 4

Apply the constant rule.

$\frac{1}{2} x^{2}]_{4}^{6} + 2 x]_{4}^{6} + \int_{4}^{6} \frac{6}{x - 2} d x$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{2} x^{2}]_{4}^{6} + 2 x]_{4}^{6} + 6 \int_{4}^{6} \frac{1}{x - 2} d x$

Step 6

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

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

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Step 6.1.1

Differentiate $x - 2$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

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

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Substitute the lower limit in for $x$ in $u = x - 2$ .

$u_{lower} = 4 - 2$

Step 6.3

Subtract $2$ from $4$ .

$u_{lower} = 2$

Step 6.4

Substitute the upper limit in for $x$ in $u = x - 2$ .

$u_{upper} = 6 - 2$

Step 6.5

Subtract $2$ from $6$ .

$u_{upper} = 4$

Step 6.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 2$

$u_{upper} = 4$

Step 6.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{1}{2} x^{2}]_{4}^{6} + 2 x]_{4}^{6} + 6 \int_{2}^{4} \frac{1}{u} d u$

Step 7

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

$\frac{1}{2} x^{2}]_{4}^{6} + 2 x]_{4}^{6} + 6 (\ln (| u |)]_{2}^{4})$

Step 8

Combine $\frac{1}{2} x^{2}]_{4}^{6}$ and $2 x]_{4}^{6}$ .

$\frac{1}{2} x^{2} + 2 x]_{4}^{6} + 6 (\ln (| u |)]_{2}^{4})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{1}{2} x^{2} + 2 x$ at $6$ and at $4$ .

$(\frac{1}{2} \cdot 6^{2} + 2 \cdot 6) - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 (\ln (| u |)]_{2}^{4})$

Step 9.2

Evaluate $\ln (| u |)$ at $4$ and at $2$ .

$(\frac{1}{2} \cdot 6^{2} + 2 \cdot 6) - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3

Simplify.

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Step 9.3.1

Raise $6$ to the power of $2$ .

$\frac{1}{2} \cdot 36 + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.2

Combine $\frac{1}{2}$ and $36$ .

$\frac{36}{2} + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.3

Cancel the common factor of $36$ and $2$ .

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Step 9.3.3.1

Factor $2$ out of $36$ .

$\frac{2 \cdot 18}{2} + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.3.2

Cancel the common factors.

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Step 9.3.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 18}{2 (1)} + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 18}{2 \cdot 1} + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.3.2.3

Rewrite the expression.

$\frac{18}{1} + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.3.2.4

Divide $18$ by $1$ .

$18 + 2 \cdot 6 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.4

Multiply $2$ by $6$ .

$18 + 12 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.5

Add $18$ and $12$ .

$30 - (\frac{1}{2} \cdot 4^{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.6

Raise $4$ to the power of $2$ .

$30 - (\frac{1}{2} \cdot 16 + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.7

Combine $\frac{1}{2}$ and $16$ .

$30 - (\frac{16}{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.8

Cancel the common factor of $16$ and $2$ .

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Step 9.3.8.1

Factor $2$ out of $16$ .

$30 - (\frac{2 \cdot 8}{2} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.8.2

Cancel the common factors.

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Step 9.3.8.2.1

Factor $2$ out of $2$ .

$30 - (\frac{2 \cdot 8}{2 (1)} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.8.2.2

Cancel the common factor.

$30 - (\frac{2 \cdot 8}{2 \cdot 1} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.8.2.3

Rewrite the expression.

$30 - (\frac{8}{1} + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.8.2.4

Divide $8$ by $1$ .

$30 - (8 + 2 \cdot 4) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.9

Multiply $2$ by $4$ .

$30 - (8 + 8) + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.10

Add $8$ and $8$ .

$30 - 1 \cdot 16 + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.11

Multiply $- 1$ by $16$ .

$30 - 16 + 6 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 9.3.12

Subtract $16$ from $30$ .

$14 + 6 (\ln (| 4 |) - \ln (| 2 |))$

Step 10

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$14 + 6 \ln (\frac{| 4 |}{| 2 |})$

Step 11

Simplify.

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Step 11.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$14 + 6 \ln (\frac{4}{| 2 |})$

Step 11.2

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$14 + 6 \ln (\frac{4}{2})$

Step 11.3

Divide $4$ by $2$ .

$14 + 6 \ln (2)$

Step 12

The result can be shown in multiple forms.

Exact Form:

$14 + 6 \ln (2)$

Decimal Form:

$18.15888308 \dots$

Step 13