Calculus Examples

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

Step 1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 2

Apply the distributive property.

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

Step 3

Apply the distributive property.

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

Step 4

Apply the distributive property.

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

Step 5

Reorder $x$ and $- 1$ .

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

Step 6

Raise $x$ to the power of $1$ .

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

Step 7

Raise $x$ to the power of $1$ .

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

Step 8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9

Simplify the expression.

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Step 9.1

Add $1$ and $1$ .

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

Step 9.2

Multiply $- 1$ by $- 1$ .

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

Step 10

Subtract $1 x$ from $- 1 x$ .

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

Step 11

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

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Step 11.1

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 11.2

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 11.3

Multiply the new quotient term by the divisor.

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

Step 11.4

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 11.5

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 11.6

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 11.7

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 11.8

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 11.9

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 11.10

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 11.11

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

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

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 14

Apply the constant rule.

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

Step 15

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 16

Simplify.

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