Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor $y$ out of $y^{3} - 2 y^{2} - y$ .

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Step 1.1.1

Factor $y$ out of $y^{3}$ .

$\int_{1}^{3} \frac{y \cdot y^{2} - 2 y^{2} - y}{y^{2}} d y$

Step 1.1.2

Factor $y$ out of $- 2 y^{2}$ .

$\int_{1}^{3} \frac{y \cdot y^{2} + y (- 2 y) - y}{y^{2}} d y$

Step 1.1.3

Factor $y$ out of $- y$ .

$\int_{1}^{3} \frac{y \cdot y^{2} + y (- 2 y) + y \cdot - 1}{y^{2}} d y$

Step 1.1.4

Factor $y$ out of $y \cdot y^{2} + y (- 2 y)$ .

$\int_{1}^{3} \frac{y (y^{2} - 2 y) + y \cdot - 1}{y^{2}} d y$

Step 1.1.5

Factor $y$ out of $y (y^{2} - 2 y) + y \cdot - 1$ .

$\int_{1}^{3} \frac{y (y^{2} - 2 y - 1)}{y^{2}} d y$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $y$ out of $y^{2}$ .

$\int_{1}^{3} \frac{y (y^{2} - 2 y - 1)}{y \cdot y} d y$

Step 1.2.2

Cancel the common factor.

$\int_{1}^{3} \frac{y (y^{2} - 2 y - 1)}{y \cdot y} d y$

Step 1.2.3

Rewrite the expression.

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

Step 2

Divide $y^{2} - 2 y - 1$ by $y$ .

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Step 2.1

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

$y$

$0$

$y^{2}$

$2 y$

$1$

Step 2.2

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

					$y$
	$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$

Step 2.3

Multiply the new quotient term by the divisor.

				$y$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			+	$y^{2}$	+	$0$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{2} + 0$

				$y$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$

Step 2.5

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

				$y$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$

Step 2.6

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

				$y$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$	-	$1$

Step 2.7

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

				$y$	-	$2$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$	-	$1$

Step 2.8

Multiply the new quotient term by the divisor.

				$y$	-	$2$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$	-	$1$
					-	$2 y$	+	$0$

Step 2.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 y + 0$

				$y$	-	$2$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$	-	$1$
					+	$2 y$	-	$0$

Step 2.10

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

				$y$	-	$2$
$y$	+	$0$		$y^{2}$	-	$2 y$	-	$1$
			-	$y^{2}$	-	$0$
					-	$2 y$	-	$1$
					+	$2 y$	-	$0$
							-	$1$

Step 2.11

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

$\frac{1}{2} y^{2} - 2 y]_{1}^{3} - (\ln (| y |)]_{1}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Evaluate $\ln (| y |)$ at $3$ and at $1$ .

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

Raise $3$ to the power of $2$ .

$\frac{1}{2} \cdot 9 - 2 \cdot 3 - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.2

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} - 2 \cdot 3 - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.3

Multiply $- 2$ by $3$ .

$\frac{9}{2} - 6 - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.4

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} - 6 \cdot \frac{2}{2} - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.5

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

$\frac{9}{2} + \frac{- 6 \cdot 2}{2} - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.6

Combine the numerators over the common denominator.

$\frac{9 - 6 \cdot 2}{2} - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.7

Simplify the numerator.

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Step 8.2.3.7.1

Multiply $- 6$ by $2$ .

$\frac{9 - 12}{2} - (\frac{1}{2} \cdot 1^{2} - 2 \cdot 1) - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.7.2

Subtract $12$ from $9$ .

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

Step 8.2.3.8

Move the negative in front of the fraction.

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

Step 8.2.3.9

One to any power is one.

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

Step 8.2.3.10

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

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

Step 8.2.3.11

Multiply $- 2$ by $1$ .

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

Step 8.2.3.12

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

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

Step 8.2.3.13

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

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

Step 8.2.3.14

Combine the numerators over the common denominator.

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

Step 8.2.3.15

Simplify the numerator.

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Step 8.2.3.15.1

Multiply $- 2$ by $2$ .

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

Step 8.2.3.15.2

Subtract $4$ from $1$ .

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

Step 8.2.3.16

Move the negative in front of the fraction.

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

Step 8.2.3.17

Multiply $- 1$ by $- 1$ .

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

Step 8.2.3.18

Multiply $\frac{3}{2}$ by $1$ .

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

Step 8.2.3.19

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

$0 - ((\ln (| 3 |)) - \ln (| 1 |))$

Step 8.2.3.20

Subtract $\ln (| 3 |) - \ln (| 1 |)$ from $0$ .

$- (\ln (| 3 |) - \ln (| 1 |))$

Step 8.3

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

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

Step 8.4

Simplify.

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Step 8.4.1

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

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

Step 8.4.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$- \ln (\frac{3}{1})$

Step 8.4.3

Divide $3$ by $1$ .

$- \ln (3)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \ln (3)$

Decimal Form:

$- 1.09861228 \dots$

Step 10