Calculus Examples

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

Step 1

Divide $x^{3} + x$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$x$

$0$

Step 1.2

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

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

Step 1.3

Multiply the new quotient term by the divisor.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Multiply the new quotient term by the divisor.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Multiply the new quotient term by the divisor.

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{3} x^{3} + C + \int - x d x + \int 2 d x + \int - \frac{2}{x + 1} d x$

Step 4

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

$\frac{1}{3} x^{3} + C - \int x d x + \int 2 d x + \int - \frac{2}{x + 1} d x$

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Multiply $2$ by $- 1$ .

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

Step 11

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 11.1.1

Differentiate $x + 1$ .

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

Step 11.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 11.1.3

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

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

Step 11.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 11.1.5

Add $1$ and $0$ .

$1$

Step 11.2

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

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

Step 12

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

$\frac{1}{3} x^{3} + C - (\frac{x^{2}}{2} + C) + 2 x + C - 2 (\ln (| u |) + C)$

Step 13

Simplify.

$\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 2 \ln (| u |) + C$

Step 14

Replace all occurrences of $u$ with $x + 1$ .

$\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 2 \ln (| x + 1 |) + C$