Calculus Examples

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

Step 1

Divide $x^{2} - 1$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$x^{2}$

$0 x$

$1$

Step 1.2

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

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

Step 1.3

Multiply the new quotient term by the divisor.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Multiply the new quotient term by the divisor.

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

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

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

Step 1.10

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

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

Step 1.11

Since the remander is $0$ , the final answer is the quotient.

$\int_{1}^{2} x - 1 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify the answer.

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Step 5.1

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

$\frac{1}{2} x^{2} - x]_{1}^{2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$\frac{1}{2} \cdot 4 - 1 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1)$

Step 5.2.2.2

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

$\frac{4}{2} - 1 \cdot 2 - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1)$

Step 5.2.2.3

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

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Step 5.2.2.3.1

Factor $2$ out of $4$ .

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

Step 5.2.2.3.2

Cancel the common factors.

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Step 5.2.2.3.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.3.2.2

Cancel the common factor.

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

Step 5.2.2.3.2.3

Rewrite the expression.

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

Step 5.2.2.3.2.4

Divide $2$ by $1$ .

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

Step 5.2.2.4

Multiply $- 1$ by $2$ .

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

Step 5.2.2.5

Subtract $2$ from $2$ .

$0 - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1)$

Step 5.2.2.6

One to any power is one.

$0 - (\frac{1}{2} \cdot 1 - 1 \cdot 1)$

Step 5.2.2.7

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

$0 - (\frac{1}{2} - 1 \cdot 1)$

Step 5.2.2.8

Multiply $- 1$ by $1$ .

$0 - (\frac{1}{2} - 1)$

Step 5.2.2.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 - (\frac{1}{2} - 1 \cdot \frac{2}{2})$

Step 5.2.2.10

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

$0 - (\frac{1}{2} + \frac{- 1 \cdot 2}{2})$

Step 5.2.2.11

Combine the numerators over the common denominator.

$0 - \frac{1 - 1 \cdot 2}{2}$

Step 5.2.2.12

Simplify the numerator.

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Step 5.2.2.12.1

Multiply $- 1$ by $2$ .

$0 - \frac{1 - 2}{2}$

Step 5.2.2.12.2

Subtract $2$ from $1$ .

$0 - \frac{- 1}{2}$

Step 5.2.2.13

Move the negative in front of the fraction.

$0 - - \frac{1}{2}$

Step 5.2.2.14

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{1}{2})$

Step 5.2.2.15

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

$0 + \frac{1}{2}$

Step 5.2.2.16

Add $0$ and $\frac{1}{2}$ .

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 7