Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Evaluate $3 x$ at $4$ and at $- 2$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $4$ to the power of $2$ .

$\frac{1}{2} (\frac{1}{2} \cdot 16 - \frac{1}{2} {(- 2)}^{2}) + 3 \cdot 4 - 3 \cdot - 2$

Step 6.3.2

Combine $\frac{1}{2}$ and $16$ .

$\frac{1}{2} (\frac{16}{2} - \frac{1}{2} {(- 2)}^{2}) + 3 \cdot 4 - 3 \cdot - 2$

Step 6.3.3

Cancel the common factor of $16$ and $2$ .

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Step 6.3.3.1

Factor $2$ out of $16$ .

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

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $2$ .

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.3.2.4

Divide $8$ by $1$ .

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

Step 6.3.4

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

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

Step 6.3.5

Multiply $4$ by $- 1$ .

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

Step 6.3.6

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

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

Step 6.3.7

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

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Step 6.3.7.1

Factor $2$ out of $- 4$ .

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

Step 6.3.7.2

Cancel the common factors.

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Step 6.3.7.2.1

Factor $2$ out of $2$ .

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

Step 6.3.7.2.2

Cancel the common factor.

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

Step 6.3.7.2.3

Rewrite the expression.

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

Step 6.3.7.2.4

Divide $- 2$ by $1$ .

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

Step 6.3.8

Subtract $2$ from $8$ .

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

Step 6.3.9

Combine $\frac{1}{2}$ and $6$ .

$\frac{6}{2} + 3 \cdot 4 - 3 \cdot - 2$

Step 6.3.10

Cancel the common factor of $6$ and $2$ .

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Step 6.3.10.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2} + 3 \cdot 4 - 3 \cdot - 2$

Step 6.3.10.2

Cancel the common factors.

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Step 6.3.10.2.1

Factor $2$ out of $2$ .

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

Step 6.3.10.2.2

Cancel the common factor.

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

Step 6.3.10.2.3

Rewrite the expression.

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

Step 6.3.10.2.4

Divide $3$ by $1$ .

$3 + 3 \cdot 4 - 3 \cdot - 2$

Step 6.3.11

Multiply $3$ by $4$ .

$3 + 12 - 3 \cdot - 2$

Step 6.3.12

Multiply $- 3$ by $- 2$ .

$3 + 12 + 6$

Step 6.3.13

Add $12$ and $6$ .

$3 + 18$

Step 6.3.14

Add $3$ and $18$ .

$21$

Step 7