Calculus Examples

$\int \frac{x^{4} - 1}{x^{2} + 1} d x$

Step 1

Divide $x^{4} - 1$ by $x^{2} + 1$ .

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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 $0$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + x^{2}$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

Step 1.5

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

Step 1.6

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0 x$

$1$

Step 1.7

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

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0 x$

$1$

Step 1.8

Multiply the new quotient term by the divisor.

						$x^{2}$	+	$0 x$	-	$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
					-	$x^{4}$	-	$0$	-	$x^{2}$
									-	$x^{2}$	+	$0 x$	-	$1$
									-	$x^{2}$	+	$0$	-	$1$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} + 0 - 1$

						$x^{2}$	+	$0 x$	-	$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
					-	$x^{4}$	-	$0$	-	$x^{2}$
									-	$x^{2}$	+	$0 x$	-	$1$
									+	$x^{2}$	-	$0$	+	$1$

Step 1.10

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

						$x^{2}$	+	$0 x$	-	$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$1$
					-	$x^{4}$	-	$0$	-	$x^{2}$
									-	$x^{2}$	+	$0 x$	-	$1$
									+	$x^{2}$	-	$0$	+	$1$
														$0$

Step 1.11

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Multiply $0$ by $x$ .

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

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int 0 d x + \int - 1 d x$

Step 5

The integral of $0$ with respect to $x$ is $0$ .

$\frac{1}{3} x^{3} + C + 0 + C + \int - 1 d x$

Step 6

Apply the constant rule.

$\frac{1}{3} x^{3} + C + 0 + C - x + C$

Step 7

Simplify.

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Step 7.1

Add $0$ and $C$ .

$\frac{1}{3} x^{3} + C + C - x + C$

Step 7.2

Simplify.

$\frac{1}{3} x^{3} - x + C$