Calculus Examples

$\int_{0}^{2} \frac{x^{2} - x - 2}{x + 1} d x$

Step 1

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

$x$

$1$

$x^{2}$

$x$

$2$

Step 1.2

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

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

Step 1.3

Multiply the new quotient term by the divisor.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Multiply the new quotient term by the divisor.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

$\int_{0}^{2} x - 2 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2} x d x + \int_{0}^{2} - 2 d x$

Step 3

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

$\frac{1}{2} x^{2}]_{0}^{2} + \int_{0}^{2} - 2 d x$

Step 4

Apply the constant rule.

$\frac{1}{2} x^{2}]_{0}^{2} + - 2 x]_{0}^{2}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2} x^{2}]_{0}^{2}$ and $- 2 x]_{0}^{2}$ .

$\frac{1}{2} x^{2} - 2 x]_{0}^{2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{2} x^{2} - 2 x$ at $2$ and at $0$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$\frac{1}{2} \cdot 4 - 2 \cdot 2 - (\frac{1}{2} \cdot 0^{2} - 2 \cdot 0)$

Step 5.2.2.2

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

$\frac{4}{2} - 2 \cdot 2 - (\frac{1}{2} \cdot 0^{2} - 2 \cdot 0)$

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 $4$ .

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

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 $2$ .

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

Step 5.2.2.3.2.2

Cancel the common factor.

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

Step 5.2.2.3.2.3

Rewrite the expression.

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

Step 5.2.2.3.2.4

Divide $2$ by $1$ .

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

Step 5.2.2.4

Multiply $- 2$ by $2$ .

$2 - 4 - (\frac{1}{2} \cdot 0^{2} - 2 \cdot 0)$

Step 5.2.2.5

Subtract $4$ from $2$ .

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

Step 5.2.2.6

Raising $0$ to any positive power yields $0$ .

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

Step 5.2.2.7

Multiply $\frac{1}{2}$ by $0$ .

$- 2 - (0 - 2 \cdot 0)$

Step 5.2.2.8

Multiply $- 2$ by $0$ .

$- 2 - (0 + 0)$

Step 5.2.2.9

Add $0$ and $0$ .

$- 2 - 0$

Step 5.2.2.10

Multiply $- 1$ by $0$ .

$- 2 + 0$

Step 5.2.2.11

Add $- 2$ and $0$ .

$- 2$

Step 6