Calculus Examples

$\frac{(x + 2) (2 x - 5)}{x}$

Step 1

Apply the distributive property.

$\int \frac{x (2 x - 5) + 2 (2 x - 5)}{x} d x$

Step 2

Apply the distributive property.

$\int \frac{x (2 x) + x \cdot - 5 + 2 (2 x - 5)}{x} d x$

Step 3

Apply the distributive property.

$\int \frac{x (2 x) + x \cdot - 5 + 2 (2 x) + 2 \cdot - 5}{x} d x$

Step 4

Simplify with commuting.

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Step 4.1

Reorder $x$ and $2$ .

$\int \frac{2 \cdot x \cdot x + x \cdot - 5 + 2 (2 x) + 2 \cdot - 5}{x} d x$

Step 4.2

Reorder $x$ and $- 5$ .

$\int \frac{2 \cdot x \cdot x - 5 \cdot x + 2 (2 x) + 2 \cdot - 5}{x} d x$

Step 5

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

$\int \frac{2 (x^{1} x) - 5 x + 2 \cdot 2 x + 2 \cdot - 5}{x} d x$

Step 6

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

$\int \frac{2 (x^{1} x^{1}) - 5 x + 2 \cdot 2 x + 2 \cdot - 5}{x} d x$

Step 7

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

$\int \frac{2 x^{1 + 1} - 5 x + 2 \cdot 2 x + 2 \cdot - 5}{x} d x$

Step 8

Simplify the expression.

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Step 8.1

Add $1$ and $1$ .

$\int \frac{2 x^{2} - 5 x + 2 \cdot 2 x + 2 \cdot - 5}{x} d x$

Step 8.2

Multiply $2$ by $2$ .

$\int \frac{2 x^{2} - 5 x + 4 x + 2 \cdot - 5}{x} d x$

Step 8.3

Multiply $2$ by $- 5$ .

$\int \frac{2 x^{2} - 5 x + 4 x - 10}{x} d x$

Step 9

Add $- 5 x$ and $4 x$ .

$\int \frac{2 x^{2} - x - 10}{x} d x$

Step 10

Divide $2 x^{2} - x - 10$ by $x$ .

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Step 10.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$2 x^{2}$

$x$

$10$

Step 10.2

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

					$2 x$
	$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$

Step 10.3

Multiply the new quotient term by the divisor.

				$2 x$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			+	$2 x^{2}$	+	$0$

Step 10.4

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

				$2 x$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$

Step 10.5

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

				$2 x$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$

Step 10.6

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

				$2 x$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$	-	$10$

Step 10.7

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

				$2 x$	-	$1$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$	-	$10$

Step 10.8

Multiply the new quotient term by the divisor.

				$2 x$	-	$1$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$	-	$10$
					-	$x$	+	$0$

Step 10.9

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

				$2 x$	-	$1$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$	-	$10$
					+	$x$	-	$0$

Step 10.10

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

				$2 x$	-	$1$
$x$	+	$0$		$2 x^{2}$	-	$x$	-	$10$
			-	$2 x^{2}$	-	$0$
					-	$x$	-	$10$
					+	$x$	-	$0$
							-	$10$

Step 10.11

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

$\int 2 x - 1 - \frac{10}{x} d x$

Step 11

Split the single integral into multiple integrals.

$\int 2 x d x + \int - 1 d x + \int - \frac{10}{x} d x$

Step 12

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x + \int - 1 d x + \int - \frac{10}{x} d x$

Step 13

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

$2 (\frac{1}{2} x^{2} + C) + \int - 1 d x + \int - \frac{10}{x} d x$

Step 14

Apply the constant rule.

$2 (\frac{1}{2} x^{2} + C) - x + C + \int - \frac{10}{x} d x$

Step 15

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{2}}{2} + C) - x + C + \int - \frac{10}{x} d x$

Step 16

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

$2 (\frac{x^{2}}{2} + C) - x + C - \int \frac{10}{x} d x$

Step 17

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$2 (\frac{x^{2}}{2} + C) - x + C - (10 \int \frac{1}{x} d x)$

Step 18

Multiply $10$ by $- 1$ .

$2 (\frac{x^{2}}{2} + C) - x + C - 10 \int \frac{1}{x} d x$

Step 19

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

$2 (\frac{x^{2}}{2} + C) - x + C - 10 (\ln (| x |) + C)$

Step 20

Simplify.

$x^{2} - x - 10 \ln (| x |) + C$