Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

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

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

Step 6.3.2

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

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Step 6.3.2.1

Factor $2$ out of $4$ .

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

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $2$ out of $2$ .

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

Step 6.3.2.2.2

Cancel the common factor.

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

Step 6.3.2.2.3

Rewrite the expression.

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

Step 6.3.2.2.4

Divide $2$ by $1$ .

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

Step 6.3.3

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

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

Step 6.3.4

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

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Step 6.3.4.1

Factor $2$ out of $16$ .

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

Step 6.3.4.2

Cancel the common factors.

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Step 6.3.4.2.1

Factor $2$ out of $2$ .

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

Step 6.3.4.2.2

Cancel the common factor.

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

Step 6.3.4.2.3

Rewrite the expression.

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

Step 6.3.4.2.4

Divide $8$ by $1$ .

$2 (2 - 1 \cdot 8) + 4 \cdot 2 - 4 \cdot - 4$

Step 6.3.5

Multiply $- 1$ by $8$ .

$2 (2 - 8) + 4 \cdot 2 - 4 \cdot - 4$

Step 6.3.6

Subtract $8$ from $2$ .

$2 \cdot - 6 + 4 \cdot 2 - 4 \cdot - 4$

Step 6.3.7

Multiply $2$ by $- 6$ .

$- 12 + 4 \cdot 2 - 4 \cdot - 4$

Step 6.3.8

Multiply $4$ by $2$ .

$- 12 + 8 - 4 \cdot - 4$

Step 6.3.9

Multiply $- 4$ by $- 4$ .

$- 12 + 8 + 16$

Step 6.3.10

Add $8$ and $16$ .

$- 12 + 24$

Step 6.3.11

Add $- 12$ and $24$ .

$12$

Step 7