Calculus Examples

$\int_{4 - 3 x}^{1} \frac{u^{3}}{1 + u^{2}} d u$

Step 1

Reorder $1$ and $u^{2}$ .

$\int_{4 - 3 x}^{1} \frac{u^{3}}{u^{2} + 1} d u$

Step 2

Divide $u^{3}$ by $u^{2} + 1$ .

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Step 2.1

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

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

Step 2.2

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

$u$

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

Step 2.3

Multiply the new quotient term by the divisor.

$u$

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

$u^{3}$

$0$

$u$

Step 2.4

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

$u$

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

$u^{3}$

$0$

$u$

Step 2.5

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

$u$

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

$u^{3}$

$0$

$u$

Step 2.6

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

$u$

$u^{2}$

$0 u$

$1$

$u^{3}$

$0 u^{2}$

$0 u$

$0$

$u^{3}$

$0$

$u$

$0$

Step 2.7

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

$\int_{4 - 3 x}^{1} u + \frac{- u}{u^{2} + 1} d u$

Step 3

Split the single integral into multiple integrals.

$\int_{4 - 3 x}^{1} u d u + \int_{4 - 3 x}^{1} \frac{- u}{u^{2} + 1} d u$

Step 4

Move the negative in front of the fraction.

$\int_{4 - 3 x}^{1} u d u + \int_{4 - 3 x}^{1} - \frac{u}{u^{2} + 1} d u$

Step 5

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

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} + \int_{4 - 3 x}^{1} - \frac{u}{u^{2} + 1} d u$

Step 6

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

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - \int_{4 - 3 x}^{1} \frac{u}{u^{2} + 1} d u$

Step 7

Let $u_{1} = u^{2} + 1$ . Then $d u_{1} = 2 u d u$ , so $\frac{1}{2} d u_{1} = u d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.1

Let $u_{1} = u^{2} + 1$ . Find $\frac{d u_{1}}{d u}$ .

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Step 7.1.1

Differentiate $u^{2} + 1$ .

$\frac{d}{d u} [u^{2} + 1]$

Step 7.1.2

By the Sum Rule, the derivative of $u^{2} + 1$ with respect to $u$ is $\frac{d}{d u} [u^{2}] + \frac{d}{d u} [1]$ .

$\frac{d}{d u} [u^{2}] + \frac{d}{d u} [1]$

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 7.1.4

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

$2 u + 0$

Step 7.1.5

Add $2 u$ and $0$ .

$2 u$

Step 7.2

Substitute the lower limit in for $u$ in $u_{1} = u^{2} + 1$ .

$u_{lower} = {(4 - 3 x)}^{2} + 1$

Step 7.3

Simplify.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Rewrite ${(4 - 3 x)}^{2}$ as $(4 - 3 x) (4 - 3 x)$ .

$u_{lower} = (4 - 3 x) (4 - 3 x) + 1$

Step 7.3.1.2

Expand $(4 - 3 x) (4 - 3 x)$ using the FOIL Method.

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Step 7.3.1.2.1

Apply the distributive property.

$u_{lower} = 4 (4 - 3 x) - 3 x (4 - 3 x) + 1$

Step 7.3.1.2.2

Apply the distributive property.

$u_{lower} = 4 \cdot 4 + 4 (- 3 x) - 3 x (4 - 3 x) + 1$

Step 7.3.1.2.3

Apply the distributive property.

$u_{lower} = 4 \cdot 4 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x) + 1$

Step 7.3.1.3

Simplify and combine like terms.

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Step 7.3.1.3.1

Simplify each term.

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Step 7.3.1.3.1.1

Multiply $4$ by $4$ .

$u_{lower} = 16 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x) + 1$

Step 7.3.1.3.1.2

Multiply $- 3$ by $4$ .

$u_{lower} = 16 - 12 x - 3 x \cdot 4 - 3 x (- 3 x) + 1$

Step 7.3.1.3.1.3

Multiply $4$ by $- 3$ .

$u_{lower} = 16 - 12 x - 12 x - 3 x (- 3 x) + 1$

Step 7.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$u_{lower} = 16 - 12 x - 12 x - 3 \cdot - 3 x \cdot x + 1$

Step 7.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 7.3.1.3.1.5.1

Move $x$ .

$u_{lower} = 16 - 12 x - 12 x - 3 \cdot - 3 (x \cdot x) + 1$

Step 7.3.1.3.1.5.2

Multiply $x$ by $x$ .

$u_{lower} = 16 - 12 x - 12 x - 3 \cdot - 3 x^{2} + 1$

Step 7.3.1.3.1.6

Multiply $- 3$ by $- 3$ .

$u_{lower} = 16 - 12 x - 12 x + 9 x^{2} + 1$

Step 7.3.1.3.2

Subtract $12 x$ from $- 12 x$ .

$u_{lower} = 16 - 24 x + 9 x^{2} + 1$

Step 7.3.2

Add $16$ and $1$ .

$u_{lower} = - 24 x + 9 x^{2} + 17$

Step 7.4

Substitute the upper limit in for $u$ in $u_{1} = u^{2} + 1$ .

$u_{upper} = 1^{2} + 1$

Step 7.5

Simplify.

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Step 7.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 7.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 24 x + 9 x^{2} + 17$

$u_{upper} = 2$

Step 7.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - \int_{- 24 x + 9 x^{2} + 17}^{2} \frac{1}{u_{1}} \cdot \frac{1}{2} d u_{1}$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{u_{1}}$ by $\frac{1}{2}$ .

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - \int_{- 24 x + 9 x^{2} + 17}^{2} \frac{1}{u_{1} \cdot 2} d u_{1}$

Step 8.2

Move $2$ to the left of $u_{1}$ .

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - \int_{- 24 x + 9 x^{2} + 17}^{2} \frac{1}{2 u_{1}} d u_{1}$

Step 9

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

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - (\frac{1}{2} \int_{- 24 x + 9 x^{2} + 17}^{2} \frac{1}{u_{1}} d u_{1})$

Step 10

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

$\frac{1}{2} u^{2}]_{4 - 3 x}^{1} - \frac{1}{2} \ln (| u_{1} |)]_{- 24 x + 9 x^{2} + 17}^{2}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{1}{2} u^{2}$ at $1$ and at $4 - 3 x$ .

$(\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} {(4 - 3 x)}^{2} - \frac{1}{2} \ln (| u_{1} |)]_{- 24 x + 9 x^{2} + 17}^{2}$

Step 11.2

Evaluate $\ln (| u_{1} |)$ at $2$ and at $- 24 x + 9 x^{2} + 17$ .

$(\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} {(4 - 3 x)}^{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3

Simplify.

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Step 11.3.1

One to any power is one.

$\frac{1}{2} \cdot 1 - \frac{1}{2} {(4 - 3 x)}^{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} - \frac{1}{2} {(4 - 3 x)}^{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.3

To write $- \frac{1}{2} {(4 - 3 x)}^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} - \frac{1}{2} {(4 - 3 x)}^{2} \cdot \frac{2}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.4

Combine $- \frac{1}{2} {(4 - 3 x)}^{2}$ and $\frac{2}{2}$ .

$\frac{1}{2} + \frac{- \frac{1}{2} {(4 - 3 x)}^{2} \cdot 2}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.5

Combine the numerators over the common denominator.

$\frac{1 - \frac{1}{2} {(4 - 3 x)}^{2} \cdot 2}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.6

Combine ${(4 - 3 x)}^{2}$ and $\frac{1}{2}$ .

$\frac{1 - \frac{{(4 - 3 x)}^{2}}{2} \cdot 2}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.7

Multiply $2$ by $- 1$ .

$\frac{1 - 2 \frac{{(4 - 3 x)}^{2}}{2}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.8

Combine $- 2$ and $\frac{{(4 - 3 x)}^{2}}{2}$ .

$\frac{1 + \frac{- 2 {(4 - 3 x)}^{2}}{2}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.9

Cancel the common factor of $- 2$ and $2$ .

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Step 11.3.9.1

Factor $2$ out of $- 2 {(4 - 3 x)}^{2}$ .

$\frac{1 + \frac{2 (- {(4 - 3 x)}^{2})}{2}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.9.2

Cancel the common factors.

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Step 11.3.9.2.1

Factor $2$ out of $2$ .

$\frac{1 + \frac{2 (- {(4 - 3 x)}^{2})}{2 (1)}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.9.2.2

Cancel the common factor.

$\frac{1 + \frac{2 (- {(4 - 3 x)}^{2})}{2 \cdot 1}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.9.2.3

Rewrite the expression.

$\frac{1 + \frac{- {(4 - 3 x)}^{2}}{1}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 11.3.9.2.4

Divide $- {(4 - 3 x)}^{2}$ by $1$ .

$\frac{1 - {(4 - 3 x)}^{2}}{2} - \frac{1}{2} ((\ln (| 2 |)) - \ln (| - 24 x + 9 x^{2} + 17 |))$

$\frac{1 - {(4 - 3 x)}^{2}}{2} - \frac{1}{2} (\ln (| 2 |) - \ln (| - 24 x + 9 x^{2} + 17 |))$

Step 12

Simplify.

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Step 12.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1 - {(4 - 3 x)}^{2}}{2} - \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$

Step 12.2

To write $- \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1 - {(4 - 3 x)}^{2}}{2} - \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}) \cdot \frac{2}{2}$

Step 12.3

Combine $- \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$ and $\frac{2}{2}$ .

$\frac{1 - {(4 - 3 x)}^{2}}{2} + \frac{- \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}) \cdot 2}{2}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{1 - {(4 - 3 x)}^{2} - \frac{1}{2} \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}) \cdot 2}{2}$

Step 12.5

Combine $\ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$ and $\frac{1}{2}$ .

$\frac{1 - {(4 - 3 x)}^{2} - \frac{\ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{2} \cdot 2}{2}$

Step 12.6

Multiply $2$ by $- 1$ .

$\frac{1 - {(4 - 3 x)}^{2} - 2 \frac{\ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{2}}{2}$

Step 12.7

Combine $- 2$ and $\frac{\ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{2}$ .

$\frac{1 - {(4 - 3 x)}^{2} + \frac{- 2 \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{2}}{2}$

Step 12.8

Cancel the common factor of $- 2$ and $2$ .

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Step 12.8.1

Factor $2$ out of $- 2 \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$ .

$\frac{1 - {(4 - 3 x)}^{2} + \frac{2 (- \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}))}{2}}{2}$

Step 12.8.2

Cancel the common factors.

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Step 12.8.2.1

Factor $2$ out of $2$ .

$\frac{1 - {(4 - 3 x)}^{2} + \frac{2 (- \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}))}{2 (1)}}{2}$

Step 12.8.2.2

Cancel the common factor.

$\frac{1 - {(4 - 3 x)}^{2} + \frac{2 (- \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}))}{2 \cdot 1}}{2}$

Step 12.8.2.3

Rewrite the expression.

$\frac{1 - {(4 - 3 x)}^{2} + \frac{- \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{1}}{2}$

Step 12.8.2.4

Divide $- \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})$ by $1$ .

$\frac{1 - {(4 - 3 x)}^{2} - \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |})}{2}$

Step 13

Reorder terms.

$\frac{1}{2} (1 - {(4 - 3 x)}^{2} - \ln (\frac{| 2 |}{| - 24 x + 9 x^{2} + 17 |}))$

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{1}{2} (1 - {(4 - 3 x)}^{2} - \ln (\frac{2}{| - 24 x + 9 x^{2} + 17 |}))$

Step 14.1.2

Reorder terms.

$\frac{1}{2} (1 - {(4 - 3 x)}^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$

Step 14.2

Apply the distributive property.

$\frac{1}{2} \cdot 1 + \frac{1}{2} (- {(4 - 3 x)}^{2}) + \frac{1}{2} (- \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$

Step 14.3

Simplify.

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Step 14.3.1

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \frac{1}{2} (- {(4 - 3 x)}^{2}) + \frac{1}{2} (- \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$

Step 14.3.2

Combine $\frac{1}{2}$ and ${(4 - 3 x)}^{2}$ .

$\frac{1}{2} - \frac{{(4 - 3 x)}^{2}}{2} + \frac{1}{2} (- \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$

Step 14.3.3

Combine $\frac{1}{2}$ and $\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})$ .

$\frac{1}{2} - \frac{{(4 - 3 x)}^{2}}{2} - \frac{\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.4

Combine the numerators over the common denominator.

$\frac{1 - {(4 - 3 x)}^{2}}{2} - \frac{\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.5

Combine the numerators over the common denominator.

$\frac{1 - {(4 - 3 x)}^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6

Simplify the numerator.

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Step 14.6.1

Rewrite ${(4 - 3 x)}^{2}$ as $(4 - 3 x) (4 - 3 x)$ .

$\frac{1 - ((4 - 3 x) (4 - 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.2

Expand $(4 - 3 x) (4 - 3 x)$ using the FOIL Method.

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Step 14.6.2.1

Apply the distributive property.

$\frac{1 - (4 (4 - 3 x) - 3 x (4 - 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.2.2

Apply the distributive property.

$\frac{1 - (4 \cdot 4 + 4 (- 3 x) - 3 x (4 - 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.2.3

Apply the distributive property.

$\frac{1 - (4 \cdot 4 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3

Simplify and combine like terms.

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Step 14.6.3.1

Simplify each term.

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Step 14.6.3.1.1

Multiply $4$ by $4$ .

$\frac{1 - (16 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.2

Multiply $- 3$ by $4$ .

$\frac{1 - (16 - 12 x - 3 x \cdot 4 - 3 x (- 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.3

Multiply $4$ by $- 3$ .

$\frac{1 - (16 - 12 x - 12 x - 3 x (- 3 x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{1 - (16 - 12 x - 12 x - 3 \cdot - 3 x \cdot x) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 14.6.3.1.5.1

Move $x$ .

$\frac{1 - (16 - 12 x - 12 x - 3 \cdot - 3 (x \cdot x)) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.5.2

Multiply $x$ by $x$ .

$\frac{1 - (16 - 12 x - 12 x - 3 \cdot - 3 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{1 - (16 - 12 x - 12 x + 9 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.3.2

Subtract $12 x$ from $- 12 x$ .

$\frac{1 - (16 - 24 x + 9 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.4

Apply the distributive property.

$\frac{1 - 1 \cdot 16 - (- 24 x) - (9 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.5

Simplify.

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Step 14.6.5.1

Multiply $- 1$ by $16$ .

$\frac{1 - 16 - (- 24 x) - (9 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.5.2

Multiply $- 24$ by $- 1$ .

$\frac{1 - 16 + 24 x - (9 x^{2}) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.5.3

Multiply $9$ by $- 1$ .

$\frac{1 - 16 + 24 x - 9 x^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.6

Subtract $16$ from $1$ .

$\frac{- 15 + 24 x - 9 x^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.7

Rewrite $- 15 + 24 x - 9 x^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})$ in a factored form.

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Step 14.6.7.1

Factor $- 1$ out of $24 x - 9 x^{2} - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})$ .

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Step 14.6.7.1.1

Reorder $24 x$ and $- 9 x^{2}$ .

$\frac{- 15 - 9 x^{2} + 24 x - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.7.1.2

Factor $- 1$ out of $- 9 x^{2}$ .

$\frac{- 15 - (9 x^{2}) + 24 x - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.7.1.3

Factor $- 1$ out of $24 x$ .

$\frac{- 15 - (9 x^{2}) - (- 24 x) - \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$

Step 14.6.7.1.4

Factor $- 1$ out of $- \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})$ .

$\frac{- 15 - (9 x^{2}) - (- 24 x) - (\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.6.7.1.5

Factor $- 1$ out of $- (9 x^{2}) - (- 24 x)$ .

$\frac{- 15 - (9 x^{2} - 24 x) - (\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.6.7.1.6

Factor $- 1$ out of $- (9 x^{2} - 24 x) - (\ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$ .

$\frac{- 15 - (9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.6.7.2

Factor $- 1$ out of $- 15 - (9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$ .

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Step 14.6.7.2.1

Rewrite $- 15$ as $- 1 (15)$ .

$\frac{- 1 (15) - (9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.6.7.2.2

Factor $- 1$ out of $- 1 (15) - (9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$ .

$\frac{- 1 (15 + 9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.6.7.2.3

Rewrite $- 1 (15 + 9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$ as $- (15 + 9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))$ .

$\frac{- (15 + 9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |}))}{2}$

Step 14.7

Move the negative in front of the fraction.

$- \frac{15 + 9 x^{2} - 24 x + \ln (\frac{2}{| 9 x^{2} - 24 x + 17 |})}{2}$