Calculus Examples

$\int \frac{\frac{1}{2} x^{4} - x^{3} - 5 x - 7}{x + 2} d x$

Step 1

Let $u = x + 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + 2$ .

$\frac{d}{d x} [x + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.1.3

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

$1 + \frac{d}{d x} [2]$

Step 1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int \frac{\frac{1}{2} {(u - 2)}^{4} - {(u - 2)}^{3} - 5 (u - 2) - 7}{u} d u$

Step 2

Split the fraction into multiple fractions.

$\int \frac{\frac{1}{2} {(u - 2)}^{4}}{u} + \frac{- {(u - 2)}^{3}}{u} + \frac{- 5 (u - 2)}{u} + \frac{- 7}{u} d u$

Step 3

Split the single integral into multiple integrals.

$\int \frac{\frac{1}{2} {(u - 2)}^{4}}{u} d u + \int \frac{- {(u - 2)}^{3}}{u} d u + \int \frac{- 5 (u - 2)}{u} d u + \int \frac{- 7}{u} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$\int \frac{\frac{1}{2} {(u - 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int \frac{- 5 (u - 2)}{u} d u + \int \frac{- 7}{u} d u$

Step 4.2

Move the negative in front of the fraction.

$\int \frac{\frac{1}{2} {(u - 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int \frac{- 7}{u} d u$

Step 5

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{{(u - 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int \frac{- 7}{u} d u$

Step 6

Move the negative in front of the fraction.

$\frac{1}{2} \int \frac{{(u - 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 7

Use the Binomial Theorem.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} {(- 2)}^{2} + 4 u {(- 2)}^{3} + {(- 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8

Simplify the expression.

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Step 8.1

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u {(- 2)}^{3} + {(- 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.2

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 {(- 2)}^{2}) + {(- 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.3

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 (- 2 \cdot - 2)) + {(- 2)}^{4}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.4

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 (- 2 \cdot - 2)) - 2 {(- 2)}^{3}}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.5

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 (- 2 \cdot - 2)) - 2 (- 2 {(- 2)}^{2})}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.6

Rewrite the exponentiation as a product.

$\frac{1}{2} \int \frac{u^{4} + 4 u^{3} \cdot - 2 + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 (- 2 \cdot - 2)) - 2 (- 2 (- 2 \cdot - 2))}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.7

Move $u^{3}$ .

$\frac{1}{2} \int \frac{u^{4} + 4 \cdot - 2 u^{3} + 6 u^{2} (- 2 \cdot - 2) + 4 u (- 2 (- 2 \cdot - 2)) - 2 (- 2 (- 2 \cdot - 2))}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.8

Move $u^{2}$ .

$\frac{1}{2} \int \frac{u^{4} + 4 \cdot - 2 u^{3} + 6 \cdot - 2 \cdot - 2 u^{2} + 4 u (- 2 (- 2 \cdot - 2)) - 2 (- 2 (- 2 \cdot - 2))}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.9

Move parentheses.

$\frac{1}{2} \int \frac{u^{4} + 4 \cdot - 2 u^{3} + 6 \cdot - 2 \cdot - 2 u^{2} + 4 u (- 2 \cdot - 2) \cdot - 2 - 2 (- 2 (- 2 \cdot - 2))}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.10

Move $u$ .

$\frac{1}{2} \int \frac{u^{4} + 4 \cdot - 2 u^{3} + 6 \cdot - 2 \cdot - 2 u^{2} + (4 (- 2 \cdot - 2)) \cdot - 2 u - 2 (- 2 (- 2 \cdot - 2))}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.11

Move parentheses.

$\frac{1}{2} \int \frac{u^{4} + 4 \cdot - 2 u^{3} + 6 \cdot - 2 \cdot - 2 u^{2} + 4 \cdot - 2 \cdot - 2 \cdot - 2 u - 2 (- 2 \cdot - 2) \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.12

Multiply $4$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 6 \cdot - 2 \cdot - 2 u^{2} + 4 \cdot - 2 \cdot - 2 \cdot - 2 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.13

Multiply $6$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} - 12 \cdot - 2 u^{2} + 4 \cdot - 2 \cdot - 2 \cdot - 2 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.14

Multiply $- 12$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} + 4 \cdot - 2 \cdot - 2 \cdot - 2 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.15

Multiply $4$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} - 8 \cdot - 2 \cdot - 2 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.16

Multiply $- 8$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} + 16 \cdot - 2 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.17

Multiply $16$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} - 32 u - 2 \cdot - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.18

Multiply $- 2$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} - 32 u + 4 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.19

Multiply $4$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} - 32 u - 8 \cdot - 2}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 8.20

Multiply $- 8$ by $- 2$ .

$\frac{1}{2} \int \frac{u^{4} - 8 u^{3} + 24 u^{2} - 32 u + 16}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 9

Divide $u^{4} - 8 u^{3} + 24 u^{2} - 32 u + 16$ by $u$ .

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Step 9.1

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

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

Step 9.2

Divide the highest order term in the dividend $u^{4}$ by the highest order term in divisor $u$ .

$u^{3}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

Step 9.3

Multiply the new quotient term by the divisor.

$u^{3}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

Step 9.4

The expression needs to be subtracted from the dividend, so change all the signs in $u^{4} + 0$

				$u^{3}$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$

Step 9.5

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

				$u^{3}$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$

Step 9.6

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

$u^{3}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

Step 9.7

Divide the highest order term in the dividend $- 8 u^{3}$ by the highest order term in divisor $u$ .

$u^{3}$

$8 u^{2}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

Step 9.8

Multiply the new quotient term by the divisor.

				$u^{3}$	-	$8 u^{2}$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					-	$8 u^{3}$	+	$0$

Step 9.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 u^{3} + 0$

				$u^{3}$	-	$8 u^{2}$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$

Step 9.10

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

$u^{3}$

$8 u^{2}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

$8 u^{3}$

$0$

$24 u^{2}$

Step 9.11

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

$u^{3}$

$8 u^{2}$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

$8 u^{3}$

$0$

$24 u^{2}$

$32 u$

Step 9.12

Divide the highest order term in the dividend $24 u^{2}$ by the highest order term in divisor $u$ .

$u^{3}$

$8 u^{2}$

$24 u$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

$8 u^{3}$

$0$

$24 u^{2}$

$32 u$

Step 9.13

Multiply the new quotient term by the divisor.

$u^{3}$

$8 u^{2}$

$24 u$

$u$

$0$

$u^{4}$

$8 u^{3}$

$24 u^{2}$

$32 u$

$16$

$u^{4}$

$0$

$8 u^{3}$

$24 u^{2}$

$8 u^{3}$

$0$

$24 u^{2}$

$32 u$

$24 u^{2}$

$0$

Step 9.14

The expression needs to be subtracted from the dividend, so change all the signs in $24 u^{2} + 0$

				$u^{3}$	-	$8 u^{2}$	+	$24 u$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$

Step 9.15

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

				$u^{3}$	-	$8 u^{2}$	+	$24 u$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$

Step 9.16

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

				$u^{3}$	-	$8 u^{2}$	+	$24 u$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$	+	$16$

Step 9.17

Divide the highest order term in the dividend $- 32 u$ by the highest order term in divisor $u$ .

				$u^{3}$	-	$8 u^{2}$	+	$24 u$	-	$32$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$	+	$16$

Step 9.18

Multiply the new quotient term by the divisor.

				$u^{3}$	-	$8 u^{2}$	+	$24 u$	-	$32$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$	+	$16$
									-	$32 u$	+	$0$

Step 9.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 32 u + 0$

				$u^{3}$	-	$8 u^{2}$	+	$24 u$	-	$32$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$	+	$16$
									+	$32 u$	-	$0$

Step 9.20

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

				$u^{3}$	-	$8 u^{2}$	+	$24 u$	-	$32$
$u$	+	$0$		$u^{4}$	-	$8 u^{3}$	+	$24 u^{2}$	-	$32 u$	+	$16$
			-	$u^{4}$	-	$0$
					-	$8 u^{3}$	+	$24 u^{2}$
					+	$8 u^{3}$	-	$0$
							+	$24 u^{2}$	-	$32 u$
							-	$24 u^{2}$	-	$0$
									-	$32 u$	+	$16$
									+	$32 u$	-	$0$
											+	$16$

Step 9.21

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

$\frac{1}{2} \int u^{3} - 8 u^{2} + 24 u - 32 + \frac{16}{u} d u + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 10

Split the single integral into multiple integrals.

$\frac{1}{2} (\int u^{3} d u + \int - 8 u^{2} d u + \int 24 u d u + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 11

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{2} (\frac{1}{4} u^{4} + C + \int - 8 u^{2} d u + \int 24 u d u + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 12

Since $- 8$ is constant with respect to $u$ , move $- 8$ out of the integral.

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 \int u^{2} d u + \int 24 u d u + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 13

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

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{1}{3} u^{3} + C) + \int 24 u d u + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 14

Since $24$ is constant with respect to $u$ , move $24$ out of the integral.

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{1}{3} u^{3} + C) + 24 \int u d u + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 15

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

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{1}{3} u^{3} + C) + 24 (\frac{1}{2} u^{2} + C) + \int - 32 d u + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 16

Apply the constant rule.

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{1}{3} u^{3} + C) + 24 (\frac{1}{2} u^{2} + C) - 32 u + C + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 17

Simplify.

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Step 17.1

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

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{1}{2} u^{2} + C) - 32 u + C + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 17.2

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

$\frac{1}{2} (\frac{1}{4} u^{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 17.3

Combine $\frac{1}{4}$ and $u^{4}$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + \int \frac{16}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 18

Since $16$ is constant with respect to $u$ , move $16$ out of the integral.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 \int \frac{1}{u} d u) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 19

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) + \int - \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 20

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{{(u - 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 21

Use the Binomial Theorem.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 u^{2} \cdot - 2 + 3 u {(- 2)}^{2} + {(- 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22

Simplify the expression.

Tap for more steps...

Step 22.1

Rewrite the exponentiation as a product.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 u^{2} \cdot - 2 + 3 u (- 2 \cdot - 2) + {(- 2)}^{3}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.2

Rewrite the exponentiation as a product.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 u^{2} \cdot - 2 + 3 u (- 2 \cdot - 2) - 2 {(- 2)}^{2}}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.3

Rewrite the exponentiation as a product.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 u^{2} \cdot - 2 + 3 u (- 2 \cdot - 2) - 2 (- 2 \cdot - 2)}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.4

Move $u^{2}$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 \cdot - 2 u^{2} + 3 u (- 2 \cdot - 2) - 2 (- 2 \cdot - 2)}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.5

Move $u$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} + 3 \cdot - 2 u^{2} + 3 \cdot - 2 \cdot - 2 u - 2 (- 2 \cdot - 2)}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.6

Multiply $3$ by $- 2$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} - 6 u^{2} + 3 \cdot - 2 \cdot - 2 u - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.7

Multiply $3$ by $- 2$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} - 6 u^{2} - 6 \cdot - 2 u - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.8

Multiply $- 6$ by $- 2$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} - 6 u^{2} + 12 u - 2 \cdot - 2 \cdot - 2}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.9

Multiply $- 2$ by $- 2$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} - 6 u^{2} + 12 u + 4 \cdot - 2}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 22.10

Multiply $4$ by $- 2$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int \frac{u^{3} - 6 u^{2} + 12 u - 8}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 23

Divide $u^{3} - 6 u^{2} + 12 u - 8$ by $u$ .

Tap for more steps...

Step 23.1

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

$u$

$0$

$u^{3}$

$6 u^{2}$

$12 u$

$8$

Step 23.2

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

					$u^{2}$
	$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$

Step 23.3

Multiply the new quotient term by the divisor.

				$u^{2}$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			+	$u^{3}$	+	$0$

Step 23.4

The expression needs to be subtracted from the dividend, so change all the signs in $u^{3} + 0$

				$u^{2}$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$

Step 23.5

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

				$u^{2}$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$

Step 23.6

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

				$u^{2}$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$

Step 23.7

Divide the highest order term in the dividend $- 6 u^{2}$ by the highest order term in divisor $u$ .

				$u^{2}$	-	$6 u$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$

Step 23.8

Multiply the new quotient term by the divisor.

				$u^{2}$	-	$6 u$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					-	$6 u^{2}$	+	$0$

Step 23.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 u^{2} + 0$

				$u^{2}$	-	$6 u$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$

Step 23.10

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

				$u^{2}$	-	$6 u$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$

Step 23.11

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

				$u^{2}$	-	$6 u$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$	-	$8$

Step 23.12

Divide the highest order term in the dividend $12 u$ by the highest order term in divisor $u$ .

				$u^{2}$	-	$6 u$	+	$12$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$	-	$8$

Step 23.13

Multiply the new quotient term by the divisor.

				$u^{2}$	-	$6 u$	+	$12$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$	-	$8$
							+	$12 u$	+	$0$

Step 23.14

The expression needs to be subtracted from the dividend, so change all the signs in $12 u + 0$

				$u^{2}$	-	$6 u$	+	$12$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$	-	$8$
							-	$12 u$	-	$0$

Step 23.15

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

				$u^{2}$	-	$6 u$	+	$12$
$u$	+	$0$		$u^{3}$	-	$6 u^{2}$	+	$12 u$	-	$8$
			-	$u^{3}$	-	$0$
					-	$6 u^{2}$	+	$12 u$
					+	$6 u^{2}$	-	$0$
							+	$12 u$	-	$8$
							-	$12 u$	-	$0$
									-	$8$

Step 23.16

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - \int u^{2} - 6 u + 12 - \frac{8}{u} d u + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 24

Split the single integral into multiple integrals.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\int u^{2} d u + \int - 6 u d u + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 25

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{1}{3} u^{3} + C + \int - 6 u d u + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 26

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C + \int - 6 u d u + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 27

Since $- 6$ is constant with respect to $u$ , move $- 6$ out of the integral.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 \int u d u + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 28

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{1}{2} u^{2} + C) + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 29

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + \int 12 d u + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 30

Apply the constant rule.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C + \int - \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 31

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - \int \frac{8}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 32

Since $8$ is constant with respect to $u$ , move $8$ out of the integral.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - (8 \int \frac{1}{u} d u)) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 33

Multiply $8$ by $- 1$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 \int \frac{1}{u} d u) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 34

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) + \int - \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 35

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - \int \frac{5 (u - 2)}{u} d u + \int - \frac{7}{u} d u$

Step 36

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - (5 \int \frac{u - 2}{u} d u) + \int - \frac{7}{u} d u$

Step 37

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 \int \frac{u - 2}{u} d u + \int - \frac{7}{u} d u$

Step 38

Divide $u - 2$ by $u$ .

Tap for more steps...

Step 38.1

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

$u$

$0$

$u$

$2$

Step 38.2

Divide the highest order term in the dividend $u$ by the highest order term in divisor $u$ .

					$1$
	$u$	+	$0$		$u$	-	$2$

Step 38.3

Multiply the new quotient term by the divisor.

				$1$
$u$	+	$0$		$u$	-	$2$
			+	$u$	+	$0$

Step 38.4

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

				$1$
$u$	+	$0$		$u$	-	$2$
			-	$u$	-	$0$

Step 38.5

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

				$1$
$u$	+	$0$		$u$	-	$2$
			-	$u$	-	$0$
					-	$2$

Step 38.6

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 \int 1 - \frac{2}{u} d u + \int - \frac{7}{u} d u$

Step 39

Split the single integral into multiple integrals.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (\int d u + \int - \frac{2}{u} d u) + \int - \frac{7}{u} d u$

Step 40

Apply the constant rule.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C + \int - \frac{2}{u} d u) + \int - \frac{7}{u} d u$

Step 41

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - \int \frac{2}{u} d u) + \int - \frac{7}{u} d u$

Step 42

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - (2 \int \frac{1}{u} d u)) + \int - \frac{7}{u} d u$

Step 43

Multiply $2$ by $- 1$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 \int \frac{1}{u} d u) + \int - \frac{7}{u} d u$

Step 44

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 (\ln (| u |) + C)) + \int - \frac{7}{u} d u$

Step 45

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 (\ln (| u |) + C)) - \int \frac{7}{u} d u$

Step 46

Since $7$ is constant with respect to $u$ , move $7$ out of the integral.

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 (\ln (| u |) + C)) - (7 \int \frac{1}{u} d u)$

Step 47

Multiply $7$ by $- 1$ .

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 (\ln (| u |) + C)) - 7 \int \frac{1}{u} d u$

Step 48

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

$\frac{1}{2} (\frac{u^{4}}{4} + C - 8 (\frac{u^{3}}{3} + C) + 24 (\frac{u^{2}}{2} + C) - 32 u + C + 16 (\ln (| u |) + C)) - (\frac{u^{3}}{3} + C - 6 (\frac{u^{2}}{2} + C) + 12 u + C - 8 (\ln (| u |) + C)) - 5 (u + C - 2 (\ln (| u |) + C)) - 7 (\ln (| u |) + C)$

Step 49

Simplify.

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Step 49.1

Simplify.

$\frac{u^{4}}{8} - \frac{4 u^{3}}{3} + 6 u^{2} - 16 u + 8 \ln (| u |) - \frac{u^{3}}{3} + 3 u^{2} - 12 u + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2

Simplify.

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Step 49.2.1

Combine the numerators over the common denominator.

$\frac{u^{4}}{8} + 6 u^{2} - 16 u + 8 \ln (| u |) + \frac{- 4 u^{3} - u^{3}}{3} + 3 u^{2} - 12 u + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.2

Subtract $u^{3}$ from $- 4 u^{3}$ .

$\frac{u^{4}}{8} + 6 u^{2} - 16 u + 8 \ln (| u |) + \frac{- 5 u^{3}}{3} + 3 u^{2} - 12 u + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.3

Move the negative in front of the fraction.

$\frac{u^{4}}{8} + 6 u^{2} - 16 u + 8 \ln (| u |) - \frac{5 u^{3}}{3} + 3 u^{2} - 12 u + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.4

Add $6 u^{2}$ and $3 u^{2}$ .

$\frac{u^{4}}{8} + 9 u^{2} - 16 u + 8 \ln (| u |) - \frac{5 u^{3}}{3} - 12 u + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.5

Subtract $12 u$ from $- 16 u$ .

$\frac{u^{4}}{8} + 9 u^{2} - 28 u + 8 \ln (| u |) - \frac{5 u^{3}}{3} + 8 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.6

Add $8 \ln (| u |)$ and $8 \ln (| u |)$ .

$\frac{u^{4}}{8} + 9 u^{2} - 28 u - \frac{5 u^{3}}{3} + 16 \ln (| u |) - 5 u + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.7

Subtract $5 u$ from $- 28 u$ .

$\frac{u^{4}}{8} + 9 u^{2} - 33 u - \frac{5 u^{3}}{3} + 16 \ln (| u |) + 10 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.8

Add $16 \ln (| u |)$ and $10 \ln (| u |)$ .

$\frac{u^{4}}{8} + 9 u^{2} - 33 u - \frac{5 u^{3}}{3} + 26 \ln (| u |) - 7 \ln (| u |) + C$

Step 49.2.9

Subtract $7 \ln (| u |)$ from $26 \ln (| u |)$ .

$\frac{u^{4}}{8} + 9 u^{2} - 33 u - \frac{5 u^{3}}{3} + 19 \ln (| u |) + C$

Step 50

Replace all occurrences of $u$ with $x + 2$ .

$\frac{{(x + 2)}^{4}}{8} + 9 {(x + 2)}^{2} - 33 (x + 2) - \frac{5 {(x + 2)}^{3}}{3} + 19 \ln (| x + 2 |) + C$

Step 51

Reorder terms.

$\frac{1}{8} {(x + 2)}^{4} + 9 {(x + 2)}^{2} - 33 (x + 2) - \frac{5}{3} {(x + 2)}^{3} + 19 \ln (| x + 2 |) + C$