Calculus Examples

$\frac{x^{3} - 2 x^{2} - x}{x^{2}}$

Step 1

Simplify.

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Step 1.1

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

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Step 1.1.1

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

$\int \frac{x \cdot x^{2} - 2 x^{2} - x}{x^{2}} d x$

Step 1.1.2

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

$\int \frac{x \cdot x^{2} + x (- 2 x) - x}{x^{2}} d x$

Step 1.1.3

Factor $x$ out of $- x$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

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

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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 $0$ .

$x$

$0$

$x^{2}$

$2 x$

$1$

Step 2.2

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

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

Step 2.3

Multiply the new quotient term by the divisor.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Multiply the new quotient term by the divisor.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

$\frac{1}{2} x^{2} + C - 2 x + C - (\ln (| x |) + C)$

Step 8

Simplify.

$\frac{1}{2} x^{2} - 2 x - \ln (| x |) + C$