Calculus Examples

$\int_{1}^{2} \frac{x^{2} - x - 5}{x + 2} d x$

Step 1

Divide $x^{2} - x - 5$ 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 $0$ .

$x$

$2$

$x^{2}$

$x$

$5$

Step 1.2

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

					$x$
	$x$	+	$2$		$x^{2}$	-	$x$	-	$5$

Step 1.3

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			+	$x^{2}$	+	$2 x$

Step 1.4

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

				$x$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$

Step 1.5

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

				$x$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$

Step 1.6

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

				$x$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$	-	$5$

Step 1.7

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$x$	-	$3$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$	-	$5$

Step 1.8

Multiply the new quotient term by the divisor.

				$x$	-	$3$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$	-	$5$
					-	$3 x$	-	$6$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 6$

				$x$	-	$3$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$	-	$5$
					+	$3 x$	+	$6$

Step 1.10

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

				$x$	-	$3$
$x$	+	$2$		$x^{2}$	-	$x$	-	$5$
			-	$x^{2}$	-	$2 x$
					-	$3 x$	-	$5$
					+	$3 x$	+	$6$
							+	$1$

Step 1.11

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{2} x^{2}]_{1}^{2} + \int_{1}^{2} - 3 d x + \int_{1}^{2} \frac{1}{x + 2} d x$

Step 4

Apply the constant rule.

$\frac{1}{2} x^{2}]_{1}^{2} + - 3 x]_{1}^{2} + \int_{1}^{2} \frac{1}{x + 2} d x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $x + 2$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

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

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

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

$u_{lower} = 1 + 2$

Step 5.3

Add $1$ and $2$ .

$u_{lower} = 3$

Step 5.4

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

$u_{upper} = 2 + 2$

Step 5.5

Add $2$ and $2$ .

$u_{upper} = 4$

Step 5.6

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

$u_{lower} = 3$

$u_{upper} = 4$

Step 5.7

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

$\frac{1}{2} x^{2}]_{1}^{2} + - 3 x]_{1}^{2} + \int_{3}^{4} \frac{1}{u} d u$

Step 6

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

$\frac{1}{2} x^{2}]_{1}^{2} + - 3 x]_{1}^{2} + \ln (| u |)]_{3}^{4}$

Step 7

Combine $\frac{1}{2} x^{2}]_{1}^{2}$ and $- 3 x]_{1}^{2}$ .

$\frac{1}{2} x^{2} - 3 x]_{1}^{2} + \ln (| u |)]_{3}^{4}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{2} x^{2} - 3 x$ at $2$ and at $1$ .

$(\frac{1}{2} \cdot 2^{2} - 3 \cdot 2) - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + \ln (| u |)]_{3}^{4}$

Step 8.2

Evaluate $\ln (| u |)$ at $4$ and at $3$ .

$(\frac{1}{2} \cdot 2^{2} - 3 \cdot 2) - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{1}{2} \cdot 4 - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.2

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.3

Cancel the common factor of $4$ and $2$ .

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Step 8.3.3.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.3.2.3

Rewrite the expression.

$\frac{2}{1} - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.3.2.4

Divide $2$ by $1$ .

$2 - 3 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.4

Multiply $- 3$ by $2$ .

$2 - 6 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.5

Subtract $6$ from $2$ .

$- 4 - (\frac{1}{2} \cdot 1^{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.6

One to any power is one.

$- 4 - (\frac{1}{2} \cdot 1 - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.7

Multiply $\frac{1}{2}$ by $1$ .

$- 4 - (\frac{1}{2} - 3 \cdot 1) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.8

Multiply $- 3$ by $1$ .

$- 4 - (\frac{1}{2} - 3) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.9

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 4 - (\frac{1}{2} - 3 \cdot \frac{2}{2}) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.10

Combine $- 3$ and $\frac{2}{2}$ .

$- 4 - (\frac{1}{2} + \frac{- 3 \cdot 2}{2}) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.11

Combine the numerators over the common denominator.

$- 4 - \frac{1 - 3 \cdot 2}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.12

Simplify the numerator.

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Step 8.3.12.1

Multiply $- 3$ by $2$ .

$- 4 - \frac{1 - 6}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.12.2

Subtract $6$ from $1$ .

$- 4 - \frac{- 5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.13

Move the negative in front of the fraction.

$- 4 - - \frac{5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.14

Multiply $- 1$ by $- 1$ .

$- 4 + 1 (\frac{5}{2}) + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.15

Multiply $\frac{5}{2}$ by $1$ .

$- 4 + \frac{5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.16

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 4 \cdot \frac{2}{2} + \frac{5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.17

Combine $- 4$ and $\frac{2}{2}$ .

$\frac{- 4 \cdot 2}{2} + \frac{5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.18

Combine the numerators over the common denominator.

$\frac{- 4 \cdot 2 + 5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.19

Simplify the numerator.

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Step 8.3.19.1

Multiply $- 4$ by $2$ .

$\frac{- 8 + 5}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.19.2

Add $- 8$ and $5$ .

$\frac{- 3}{2} + (\ln (| 4 |)) - \ln (| 3 |)$

Step 8.3.20

Move the negative in front of the fraction.

$- \frac{3}{2} + \ln (| 4 |) - \ln (| 3 |)$

Step 9

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

$- \frac{3}{2} + \ln (\frac{| 4 |}{| 3 |})$

Step 10

Simplify.

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Step 10.1

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

$- \frac{3}{2} + \ln (\frac{4}{| 3 |})$

Step 10.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- \frac{3}{2} + \ln (\frac{4}{3})$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2} + \ln (\frac{4}{3})$

Decimal Form:

$- 1.21231792 \dots$

Step 12