Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

Simplify.

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Step 7.1.3.1

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

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

Step 7.1.3.2

Multiply $4$ by $3$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + \frac{12}{2} - 4 (\frac{1}{2})$

Step 7.1.3.3

Cancel the common factor of $12$ and $2$ .

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Step 7.1.3.3.1

Factor $2$ out of $12$ .

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

Step 7.1.3.3.2

Cancel the common factors.

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Step 7.1.3.3.2.1

Factor $2$ out of $2$ .

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

Step 7.1.3.3.2.2

Cancel the common factor.

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

Step 7.1.3.3.2.3

Rewrite the expression.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + \frac{6}{1} - 4 (\frac{1}{2})$

Step 7.1.3.3.2.4

Divide $6$ by $1$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 - 4 (\frac{1}{2})$

Step 7.1.3.4

Combine $- 4$ and $\frac{1}{2}$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 + \frac{- 4}{2}$

Step 7.1.3.5

Cancel the common factor of $- 4$ and $2$ .

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Step 7.1.3.5.1

Factor $2$ out of $- 4$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 + \frac{2 \cdot - 2}{2}$

Step 7.1.3.5.2

Cancel the common factors.

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Step 7.1.3.5.2.1

Factor $2$ out of $2$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 + \frac{2 \cdot - 2}{2 (1)}$

Step 7.1.3.5.2.2

Cancel the common factor.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 + \frac{2 \cdot - 2}{2 \cdot 1}$

Step 7.1.3.5.2.3

Rewrite the expression.

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 + \frac{- 2}{1}$

Step 7.1.3.5.2.4

Divide $- 2$ by $1$ .

$- 2 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{{(\frac{1}{2})}^{2}}{2}) + 6 - 2$

Step 7.1.3.6

Subtract $2$ from $6$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Combine the numerators over the common denominator.

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

Step 7.2.1.2

Simplify each term.

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Step 7.2.1.2.1

Apply the product rule to $\frac{3}{2}$ .

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

Step 7.2.1.2.2

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

$- 2 \frac{\frac{9}{2^{2}} - {(\frac{1}{2})}^{2}}{2} + 4$

Step 7.2.1.2.3

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

$- 2 \frac{\frac{9}{4} - {(\frac{1}{2})}^{2}}{2} + 4$

Step 7.2.1.2.4

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

$- 2 \frac{\frac{9}{4} - \frac{1^{2}}{2^{2}}}{2} + 4$

Step 7.2.1.2.5

One to any power is one.

$- 2 \frac{\frac{9}{4} - \frac{1}{2^{2}}}{2} + 4$

Step 7.2.1.2.6

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

$- 2 \frac{\frac{9}{4} - \frac{1}{4}}{2} + 4$

Step 7.2.1.3

Combine the numerators over the common denominator.

$- 2 \frac{\frac{9 - 1}{4}}{2} + 4$

Step 7.2.1.4

Subtract $1$ from $9$ .

$- 2 \frac{\frac{8}{4}}{2} + 4$

Step 7.2.1.5

Divide $8$ by $4$ .

$- 2 (\frac{2}{2}) + 4$

Step 7.2.1.6

Cancel the common factor of $2$ .

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Step 7.2.1.6.1

Cancel the common factor.

$- 2 (\frac{2}{2}) + 4$

Step 7.2.1.6.2

Rewrite the expression.

$- 2 \cdot 1 + 4$

Step 7.2.1.7

Multiply $- 2$ by $1$ .

$- 2 + 4$

Step 7.2.2

Add $- 2$ and $4$ .

$2$

Step 8