Calculus Examples

$\int_{0}^{8} {(\frac{2 - 3 x}{4})}^{2} d x$

Step 1

Let $u = \frac{2 - 3 x}{4}$ . Then $d u = - \frac{3}{4} d x$ , so $- \frac{4}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{2 - 3 x}{4}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{2 - 3 x}{4}$ .

$\frac{d}{d x} [\frac{2 - 3 x}{4}]$

Step 1.1.2

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{2 - 3 x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [2 - 3 x]$ .

$\frac{1}{4} \frac{d}{d x} [2 - 3 x]$

Step 1.1.3

By the Sum Rule, the derivative of $2 - 3 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 3 x]$ .

$\frac{1}{4} (\frac{d}{d x} [2] + \frac{d}{d x} [- 3 x])$

Step 1.1.4

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

$\frac{1}{4} (0 + \frac{d}{d x} [- 3 x])$

Step 1.1.5

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{1}{4} \frac{d}{d x} [- 3 x]$

Step 1.1.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{1}{4} (- 3 \frac{d}{d x} [x])$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{- 3}{4} \frac{d}{d x} [x]$

Step 1.1.7.2

Move the negative in front of the fraction.

$- \frac{3}{4} \frac{d}{d x} [x]$

Step 1.1.8

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

$- \frac{3}{4} \cdot 1$

Step 1.1.9

Multiply $- 1$ by $1$ .

$- \frac{3}{4}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{2 - 3 x}{4}$ .

$u_{lower} = \frac{2 - 3 \cdot 0}{4}$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $2 - 3 \cdot 0$ and $4$ .

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Step 1.3.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$u_{lower} = \frac{- 1 (- 2) - 3 \cdot 0}{4}$

Step 1.3.1.2

Factor $- 1$ out of $- 3 \cdot 0$ .

$u_{lower} = \frac{- 1 (- 2) - (3 \cdot 0)}{4}$

Step 1.3.1.3

Factor $- 1$ out of $- 1 (- 2) - (3 \cdot 0)$ .

$u_{lower} = \frac{- 1 (- 2 + 3 \cdot 0)}{4}$

Step 1.3.1.4

Reorder terms.

$u_{lower} = \frac{- 1 (- 2 + 0 \cdot 3)}{4}$

Step 1.3.1.5

Factor $2$ out of $- 1 (- 2 + 0 \cdot 3)$ .

$u_{lower} = \frac{2 (- 1 (- 1 + 0 \cdot 3))}{4}$

Step 1.3.1.6

Cancel the common factors.

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Step 1.3.1.6.1

Factor $2$ out of $4$ .

$u_{lower} = \frac{2 (- 1 (- 1 + 0 \cdot 3))}{2 (2)}$

Step 1.3.1.6.2

Cancel the common factor.

$u_{lower} = \frac{2 (- 1 (- 1 + 0 \cdot 3))}{2 \cdot 2}$

Step 1.3.1.6.3

Rewrite the expression.

$u_{lower} = \frac{- 1 (- 1 + 0 \cdot 3)}{2}$

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Multiply $0$ by $3$ .

$u_{lower} = \frac{- 1 (- 1 + 0)}{2}$

Step 1.3.2.2

Add $- 1$ and $0$ .

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

Step 1.3.3

Multiply $- 1$ by $- 1$ .

$u_{lower} = \frac{1}{2}$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{2 - 3 x}{4}$ .

$u_{upper} = \frac{2 - 3 \cdot 8}{4}$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2 - 3 \cdot 8$ and $4$ .

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Step 1.5.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$u_{upper} = \frac{- 1 (- 2) - 3 \cdot 8}{4}$

Step 1.5.1.2

Factor $- 1$ out of $- 3 \cdot 8$ .

$u_{upper} = \frac{- 1 (- 2) - (3 \cdot 8)}{4}$

Step 1.5.1.3

Factor $- 1$ out of $- 1 (- 2) - (3 \cdot 8)$ .

$u_{upper} = \frac{- 1 (- 2 + 3 \cdot 8)}{4}$

Step 1.5.1.4

Reorder terms.

$u_{upper} = \frac{- 1 (3 \cdot 8 - 2)}{4}$

Step 1.5.1.5

Factor $2$ out of $- 1 (3 \cdot 8 - 2)$ .

$u_{upper} = \frac{2 (- 1 (3 \cdot 4 - 1))}{4}$

Step 1.5.1.6

Cancel the common factors.

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Step 1.5.1.6.1

Factor $2$ out of $4$ .

$u_{upper} = \frac{2 (- 1 (3 \cdot 4 - 1))}{2 (2)}$

Step 1.5.1.6.2

Cancel the common factor.

$u_{upper} = \frac{2 (- 1 (3 \cdot 4 - 1))}{2 \cdot 2}$

Step 1.5.1.6.3

Rewrite the expression.

$u_{upper} = \frac{- 1 (3 \cdot 4 - 1)}{2}$

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

Multiply $3$ by $4$ .

$u_{upper} = \frac{- 1 (12 - 1)}{2}$

Step 1.5.2.2

Subtract $1$ from $12$ .

$u_{upper} = \frac{- 1 \cdot 11}{2}$

Step 1.5.3

Multiply $- 1$ by $11$ .

$u_{upper} = \frac{- 11}{2}$

Step 1.5.4

Move the negative in front of the fraction.

$u_{upper} = - \frac{11}{2}$

Step 1.6

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

$u_{lower} = \frac{1}{2}$

$u_{upper} = - \frac{11}{2}$

Step 1.7

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

$\int_{\frac{1}{2}}^{- \frac{11}{2}} - u^{2} \frac{- 1}{- \frac{3}{4}} d u$

Step 2

Simplify.

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Step 2.1

Dividing two negative values results in a positive value.

$\int_{\frac{1}{2}}^{- \frac{11}{2}} - u^{2} \frac{1}{\frac{3}{4}} d u$

Step 2.2

Multiply by the reciprocal of the fraction to divide by $\frac{3}{4}$ .

$\int_{\frac{1}{2}}^{- \frac{11}{2}} - u^{2} (1 (\frac{4}{3})) d u$

Step 2.3

Multiply $\frac{4}{3}$ by $1$ .

$\int_{\frac{1}{2}}^{- \frac{11}{2}} - u^{2} \frac{4}{3} d u$

Step 2.4

Combine $\frac{4}{3}$ and $u^{2}$ .

$\int_{\frac{1}{2}}^{- \frac{11}{2}} - \frac{4 u^{2}}{3} d u$

Step 3

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

$- \int_{\frac{1}{2}}^{- \frac{11}{2}} \frac{4 u^{2}}{3} d u$

Step 4

Since $\frac{4}{3}$ is constant with respect to $u$ , move $\frac{4}{3}$ out of the integral.

$- (\frac{4}{3} \int_{\frac{1}{2}}^{- \frac{11}{2}} u^{2} d u)$

Step 5

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

$- \frac{4}{3} \frac{1}{3} u^{3}]_{\frac{1}{2}}^{- \frac{11}{2}}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{1}{3} u^{3}$ at $- \frac{11}{2}$ and at $\frac{1}{2}$ .

$- \frac{4}{3} ((\frac{1}{3} {(- \frac{11}{2})}^{3}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 6.2

Simplify.

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Step 6.2.1

Factor $- 1$ out of $- \frac{11}{2}$ .

$- \frac{4}{3} (\frac{1}{3} {(- (\frac{11}{2}))}^{3} - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 6.2.2

Apply the product rule to $- (\frac{11}{2})$ .

$- \frac{4}{3} (\frac{1}{3} ({(- 1)}^{3} {(\frac{11}{2})}^{3}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 6.2.3

Raise $- 1$ to the power of $3$ .

$- \frac{4}{3} (\frac{1}{3} (- {(\frac{11}{2})}^{3}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the product rule to $\frac{11}{2}$ .

$- \frac{4}{3} (\frac{1}{3} (- \frac{11^{3}}{2^{3}}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7.1.2

Raise $11$ to the power of $3$ .

$- \frac{4}{3} (\frac{1}{3} (- \frac{1331}{2^{3}}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7.1.3

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

$- \frac{4}{3} (\frac{1}{3} (- \frac{1331}{8}) - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7.1.4

Multiply $\frac{1}{3} (- \frac{1331}{8})$ .

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Step 7.1.4.1

Multiply $\frac{1}{3}$ by $\frac{1331}{8}$ .

$- \frac{4}{3} (- \frac{1331}{3 \cdot 8} - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7.1.4.2

Multiply $3$ by $8$ .

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{3} {(\frac{1}{2})}^{3})$

Step 7.1.5

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

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{3} \cdot \frac{1^{3}}{2^{3}})$

Step 7.1.6

One to any power is one.

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{3} \cdot \frac{1}{2^{3}})$

Step 7.1.7

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

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{3} \cdot \frac{1}{8})$

Step 7.1.8

Multiply $- \frac{1}{3} \cdot \frac{1}{8}$ .

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Step 7.1.8.1

Multiply $\frac{1}{8}$ by $\frac{1}{3}$ .

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{8 \cdot 3})$

Step 7.1.8.2

Multiply $8$ by $3$ .

$- \frac{4}{3} (- \frac{1331}{24} - \frac{1}{24})$

Step 7.2

Combine the numerators over the common denominator.

$- \frac{4}{3} \cdot \frac{- 1331 - 1}{24}$

Step 7.3

Subtract $1$ from $- 1331$ .

$- \frac{4}{3} \cdot \frac{- 1332}{24}$

Step 7.4

Cancel the common factor of $4$ .

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Step 7.4.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 1332}{24}$

Step 7.4.2

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 1332}{24}$

Step 7.4.3

Factor $4$ out of $24$ .

$\frac{4 \cdot - 1}{3} \cdot \frac{- 1332}{4 \cdot 6}$

Step 7.4.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 1332}{4 \cdot 6}$

Step 7.4.5

Rewrite the expression.

$\frac{- 1}{3} \cdot \frac{- 1332}{6}$

Step 7.5

Cancel the common factor of $3$ .

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Step 7.5.1

Factor $3$ out of $- 1332$ .

$\frac{- 1}{3} \cdot \frac{3 (- 444)}{6}$

Step 7.5.2

Cancel the common factor.

$\frac{- 1}{3} \cdot \frac{3 \cdot - 444}{6}$

Step 7.5.3

Rewrite the expression.

$- \frac{- 444}{6}$

Step 7.6

Divide $- 444$ by $6$ .

$- - 74$

Step 7.7

Multiply $- 1$ by $- 74$ .

$74$

Step 8