Calculus Examples

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

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.

$\int x - 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int x d x + \int - 1 d x$

Step 3

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

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

Step 4

Apply the constant rule.

$\frac{1}{2} x^{2} + C - x + C$

Step 5

Simplify.

$\frac{1}{2} x^{2} - x + C$