Calculus Examples

$\int_{0}^{\frac{1}{2}} \frac{x^{4}}{x - 1} d x$

Step 1

Divide $x^{4}$ 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 $0$ .

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 1.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

Step 1.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

Step 1.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.7

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.9

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.10

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.11

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 1.12

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 1.13

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.14

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.15

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.16

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$0$

Step 1.17

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$0$

Step 1.18

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$0$

$x$

$1$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$0$

$x$

$1$

Step 1.20

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$0$

$x$

$1$

Step 1.21

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

$\int_{0}^{\frac{1}{2}} x^{3} + x^{2} + x + 1 + \frac{1}{x - 1} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{\frac{1}{2}} x^{3} d x + \int_{0}^{\frac{1}{2}} x^{2} d x + \int_{0}^{\frac{1}{2}} x d x + \int_{0}^{\frac{1}{2}} d x + \int_{0}^{\frac{1}{2}} \frac{1}{x - 1} d x$

Step 3

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

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \int_{0}^{\frac{1}{2}} x^{2} d x + \int_{0}^{\frac{1}{2}} x d x + \int_{0}^{\frac{1}{2}} d x + \int_{0}^{\frac{1}{2}} \frac{1}{x - 1} d x$

Step 4

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

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \int_{0}^{\frac{1}{2}} x d x + \int_{0}^{\frac{1}{2}} d x + \int_{0}^{\frac{1}{2}} \frac{1}{x - 1} d x$

Step 5

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

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + \int_{0}^{\frac{1}{2}} d x + \int_{0}^{\frac{1}{2}} \frac{1}{x - 1} d x$

Step 6

Apply the constant rule.

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + x]_{0}^{\frac{1}{2}} + \int_{0}^{\frac{1}{2}} \frac{1}{x - 1} d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $x - 1$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

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

$1 + 0$

Step 7.1.5

Add $1$ and $0$ .

$1$

Step 7.2

Substitute the lower limit in for $x$ in $u = x - 1$ .

$u_{lower} = 0 - 1$

Step 7.3

Subtract $1$ from $0$ .

$u_{lower} = - 1$

Step 7.4

Substitute the upper limit in for $x$ in $u = x - 1$ .

$u_{upper} = \frac{1}{2} - 1$

Step 7.5

Simplify.

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Step 7.5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$u_{upper} = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 7.5.2

Combine $- 1$ and $\frac{2}{2}$ .

$u_{upper} = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 7.5.3

Combine the numerators over the common denominator.

$u_{upper} = \frac{1 - 1 \cdot 2}{2}$

Step 7.5.4

Simplify the numerator.

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Step 7.5.4.1

Multiply $- 1$ by $2$ .

$u_{upper} = \frac{1 - 2}{2}$

Step 7.5.4.2

Subtract $2$ from $1$ .

$u_{upper} = \frac{- 1}{2}$

Step 7.5.5

Move the negative in front of the fraction.

$u_{upper} = - \frac{1}{2}$

Step 7.6

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

$u_{lower} = - 1$

$u_{upper} = - \frac{1}{2}$

Step 7.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + x]_{0}^{\frac{1}{2}} + \int_{- 1}^{- \frac{1}{2}} \frac{1}{u} d u$

Step 8

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

$\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + x]_{0}^{\frac{1}{2}} + \ln (| u |)]_{- 1}^{- \frac{1}{2}}$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{4} x^{4}]_{0}^{\frac{1}{2}}$ and $\frac{1}{3} x^{3}]_{0}^{\frac{1}{2}}$ .

$\frac{1}{4} x^{4} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + x]_{0}^{\frac{1}{2}} + \ln (| u |)]_{- 1}^{- \frac{1}{2}}$

Step 9.2

Combine $\frac{1}{4} x^{4} + \frac{1}{3} x^{3}]_{0}^{\frac{1}{2}}$ and $\frac{1}{2} x^{2}]_{0}^{\frac{1}{2}}$ .

$\frac{1}{4} x^{4} + \frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}} + x]_{0}^{\frac{1}{2}} + \ln (| u |)]_{- 1}^{- \frac{1}{2}}$

Step 9.3

Combine $\frac{1}{4} x^{4} + \frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{\frac{1}{2}}$ and $x]_{0}^{\frac{1}{2}}$ .

$\frac{1}{4} x^{4} + \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x]_{0}^{\frac{1}{2}} + \ln (| u |)]_{- 1}^{- \frac{1}{2}}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{1}{4} x^{4} + \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x$ at $\frac{1}{2}$ and at $0$ .

$(\frac{1}{4} {(\frac{1}{2})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2}) - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + \ln (| u |)]_{- 1}^{- \frac{1}{2}}$

Step 10.2

Evaluate $\ln (| u |)$ at $- \frac{1}{2}$ and at $- 1$ .

Step 10.3

Simplify.

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Step 10.3.1

Rewrite $\frac{1}{4}$ as $4^{- 1}$ .

$4^{- 1} {(\frac{1}{2})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.2

Rewrite $4$ as $2^{2}$ .

${(2^{2})}^{- 1} {(\frac{1}{2})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{2 \cdot - 1} {(\frac{1}{2})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.4

Multiply $2$ by $- 1$ .

$2^{- 2} {(\frac{1}{2})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.5

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

$2^{- 2} {(2^{- 1})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.6

Raise $2$ to the power of $1$ .

$2^{- 2} {({(2^{1})}^{- 1})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{- 2} {(2^{1 \cdot - 1})}^{4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.8

Multiply $- 1$ by $1$ .

Step 10.3.9

Multiply the exponents in ${(2^{- 1})}^{4}$ .

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Step 10.3.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{- 2} \cdot 2^{- 1 \cdot 4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.9.2

Multiply $- 1$ by $4$ .

$2^{- 2} \cdot 2^{- 4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.10

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

$2^{- 2 - 4} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.11

Subtract $4$ from $- 2$ .

$2^{- 6} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.12

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2^{6}} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.13

Raise $2$ to the power of $6$ .

$\frac{1}{64} + \frac{1}{3} {(\frac{1}{2})}^{3} + \frac{1}{2} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.14

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{2}$ by adding the exponents.

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Step 10.3.14.1

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{2}$ .

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Step 10.3.14.1.1

Raise $\frac{1}{2}$ to the power of $1$ .

$\frac{1}{64} + \frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{1} {(\frac{1}{2})}^{2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.14.1.2

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

$\frac{1}{64} + \frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{1 + 2} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.14.2

Add $1$ and $2$ .

$\frac{1}{64} + \frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{1}{2} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.15

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{1}{64} + \frac{1}{2} \cdot \frac{32}{32} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.16

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.16.1

Multiply $\frac{1}{2}$ by $\frac{32}{32}$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{1}{64} + \frac{32}{2 \cdot 32} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.16.2

Multiply $2$ by $32$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{1}{64} + \frac{32}{64} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.17

Combine the numerators over the common denominator.

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{1 + 32}{64} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.18

Add $1$ and $32$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (\frac{1}{4} \cdot 0^{4} + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.19

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (\frac{1}{4} \cdot 0 + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.20

Multiply $\frac{1}{4}$ by $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + \frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.21

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + \frac{1}{3} \cdot 0 + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.22

Multiply $\frac{1}{3}$ by $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + 0 + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.23

Add $0$ and $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + \frac{1}{2} \cdot 0^{2} + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.24

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + \frac{1}{2} \cdot 0 + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.25

Multiply $\frac{1}{2}$ by $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + 0 + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.26

Add $0$ and $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - (0 + 0) + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.27

Add $0$ and $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} - 0 + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.28

Multiply $- 1$ by $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} + 0 + (\ln (| - \frac{1}{2} |)) - \ln (| - 1 |)$

Step 10.3.29

Add $\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64}$ and $0$ .

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (| - \frac{1}{2} |) - \ln (| - 1 |)$

Step 11

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

$\frac{1}{3} {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12

Simplify.

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Step 12.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{3} \cdot \frac{1^{3}}{2^{3}} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.2

Combine.

$\frac{1 \cdot 1^{3}}{3 \cdot 2^{3}} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.3

Multiply $1^{3}$ by $1$ .

$\frac{1^{3}}{3 \cdot 2^{3}} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.4

Raise $2$ to the power of $3$ .

$\frac{1^{3}}{3 \cdot 8} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.5

Multiply $3$ by $8$ .

$\frac{1^{3}}{24} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.6

One to any power is one.

$\frac{1}{24} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{| - \frac{1}{2} |}{| - 1 |})$

Step 12.7

$- \frac{1}{2}$ is approximately $- 0.5$ which is negative so negate $- \frac{1}{2}$ and remove the absolute value

$\frac{1}{24} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{\frac{1}{2}}{| - 1 |})$

Step 12.8

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$\frac{1}{24} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{\frac{1}{2}}{1})$

Step 12.9

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

$\frac{1}{24} + {(\frac{1}{2})}^{3} + \frac{33}{64} + \ln (\frac{1}{2})$

Step 12.10

To write $\frac{1}{24}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

${(\frac{1}{2})}^{3} + \frac{1}{24} \cdot \frac{8}{8} + \frac{33}{64} + \ln (\frac{1}{2})$

Step 12.11

To write $\frac{33}{64}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{1}{2})}^{3} + \frac{1}{24} \cdot \frac{8}{8} + \frac{33}{64} \cdot \frac{3}{3} + \ln (\frac{1}{2})$

Step 12.12

Write each expression with a common denominator of $192$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.12.1

Multiply $\frac{1}{24}$ by $\frac{8}{8}$ .

${(\frac{1}{2})}^{3} + \frac{8}{24 \cdot 8} + \frac{33}{64} \cdot \frac{3}{3} + \ln (\frac{1}{2})$

Step 12.12.2

Multiply $24$ by $8$ .

${(\frac{1}{2})}^{3} + \frac{8}{192} + \frac{33}{64} \cdot \frac{3}{3} + \ln (\frac{1}{2})$

Step 12.12.3

Multiply $\frac{33}{64}$ by $\frac{3}{3}$ .

${(\frac{1}{2})}^{3} + \frac{8}{192} + \frac{33 \cdot 3}{64 \cdot 3} + \ln (\frac{1}{2})$

Step 12.12.4

Multiply $64$ by $3$ .

${(\frac{1}{2})}^{3} + \frac{8}{192} + \frac{33 \cdot 3}{192} + \ln (\frac{1}{2})$

Step 12.13

Combine the numerators over the common denominator.

${(\frac{1}{2})}^{3} + \frac{8 + 33 \cdot 3}{192} + \ln (\frac{1}{2})$

Step 12.14

Simplify the numerator.

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Step 12.14.1

Multiply $33$ by $3$ .

${(\frac{1}{2})}^{3} + \frac{8 + 99}{192} + \ln (\frac{1}{2})$

Step 12.14.2

Add $8$ and $99$ .

${(\frac{1}{2})}^{3} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{3}}{2^{3}} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.1.2

One to any power is one.

$\frac{1}{2^{3}} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.1.3

Raise $2$ to the power of $3$ .

$\frac{1}{8} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.2

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$\frac{1}{8} \cdot \frac{24}{24} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.3

Write each expression with a common denominator of $192$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.3.1

Multiply $\frac{1}{8}$ by $\frac{24}{24}$ .

$\frac{24}{8 \cdot 24} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.3.2

Multiply $8$ by $24$ .

$\frac{24}{192} + \frac{107}{192} + \ln (\frac{1}{2})$

Step 13.4

Combine the numerators over the common denominator.

$\frac{24 + 107}{192} + \ln (\frac{1}{2})$

Step 13.5

Add $24$ and $107$ .

$\frac{131}{192} + \ln (\frac{1}{2})$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{131}{192} + \ln (\frac{1}{2})$

Decimal Form:

$- 0.01085551 \dots$

Step 15