Calculus Examples

$\int \frac{2 x^{3} + 4 x^{2} - 5}{x + 3} d x$

Step 1

Divide $2 x^{3} + 4 x^{2} - 5$ by $x + 3$ .

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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 $0$ .

$x$

$3$

$2 x^{3}$

$4 x^{2}$

$0 x$

$5$

Step 1.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$

Step 1.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			+	$2 x^{3}$	+	$6 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 6 x^{2}$

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$

Step 1.5

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

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$

Step 1.6

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

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$

Step 1.7

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$

Step 1.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					-	$2 x^{2}$	-	$6 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 6 x$

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$

Step 1.10

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$

Step 1.11

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$	-	$5$

Step 1.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$	+	$6$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$	-	$5$

Step 1.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	+	$6$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$	-	$5$
							+	$6 x$	+	$18$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 18$

				$2 x^{2}$	-	$2 x$	+	$6$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$	-	$5$
							-	$6 x$	-	$18$

Step 1.15

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

				$2 x^{2}$	-	$2 x$	+	$6$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	+	$6 x$
							+	$6 x$	-	$5$
							-	$6 x$	-	$18$
									-	$23$

Step 1.16

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

$\int 2 x^{2} - 2 x + 6 - \frac{23}{x + 3} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int - 2 x d x + \int 6 d x + \int - \frac{23}{x + 3} d x$

Step 3

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

$2 \int x^{2} d x + \int - 2 x d x + \int 6 d x + \int - \frac{23}{x + 3} d x$

Step 4

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

$2 (\frac{1}{3} x^{3} + C) + \int - 2 x d x + \int 6 d x + \int - \frac{23}{x + 3} d x$

Step 5

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

$2 (\frac{1}{3} x^{3} + C) - 2 \int x d x + \int 6 d x + \int - \frac{23}{x + 3} d x$

Step 6

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

$2 (\frac{1}{3} x^{3} + C) - 2 (\frac{1}{2} x^{2} + C) + \int 6 d x + \int - \frac{23}{x + 3} d x$

Step 7

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) - 2 (\frac{1}{2} x^{2} + C) + 6 x + C + \int - \frac{23}{x + 3} d x$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{1}{2} x^{2} + C) + 6 x + C + \int - \frac{23}{x + 3} d x$

Step 8.2

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

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C + \int - \frac{23}{x + 3} d x$

Step 9

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

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C - \int \frac{23}{x + 3} d x$

Step 10

Since $23$ is constant with respect to $x$ , move $23$ out of the integral.

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C - (23 \int \frac{1}{x + 3} d x)$

Step 11

Multiply $23$ by $- 1$ .

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C - 23 \int \frac{1}{x + 3} d x$

Step 12

Let $u = x + 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 12.1

Let $u = x + 3$ . Find $\frac{d u}{d x}$ .

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Step 12.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 12.1.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 12.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [3]$

Step 12.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1 + 0$

Step 12.1.5

Add $1$ and $0$ .

$1$

Step 12.2

Rewrite the problem using $u$ and $d u$ .

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C - 23 \int \frac{1}{u} d u$

Step 13

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

$2 (\frac{x^{3}}{3} + C) - 2 (\frac{x^{2}}{2} + C) + 6 x + C - 23 (\ln (| u |) + C)$

Step 14

Simplify.

$\frac{2 x^{3}}{3} - x^{2} + 6 x - 23 \ln (| u |) + C$

Step 15

Replace all occurrences of $u$ with $x + 3$ .

$\frac{2 x^{3}}{3} - x^{2} + 6 x - 23 \ln (| x + 3 |) + C$

Step 16

Reorder terms.

$\frac{2}{3} x^{3} - x^{2} + 6 x - 23 \ln (| x + 3 |) + C$