Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

Apply the constant rule.

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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Step 4.2.2.3.1

Factor $2$ out of $4$ .

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

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.3.2.2

Cancel the common factor.

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

Step 4.2.2.3.2.3

Rewrite the expression.

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

Step 4.2.2.3.2.4

Divide $2$ by $1$ .

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

Step 4.2.2.4

Add $2$ and $2$ .

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

Step 4.2.2.5

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

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

Step 4.2.2.6

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

$4 - (\frac{1}{2} - 1)$

Step 4.2.2.7

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

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

Step 4.2.2.8

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

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

Step 4.2.2.9

Combine the numerators over the common denominator.

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

Step 4.2.2.10

Simplify the numerator.

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Step 4.2.2.10.1

Multiply $- 1$ by $2$ .

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

Step 4.2.2.10.2

Subtract $2$ from $1$ .

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

Step 4.2.2.11

Move the negative in front of the fraction.

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

Step 4.2.2.12

Multiply $- 1$ by $- 1$ .

$4 + 1 (\frac{1}{2})$

Step 4.2.2.13

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

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

Step 4.2.2.14

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.2.15

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

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

Step 4.2.2.16

Combine the numerators over the common denominator.

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

Step 4.2.2.17

Simplify the numerator.

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Step 4.2.2.17.1

Multiply $4$ by $2$ .

$\frac{8 + 1}{2}$

Step 4.2.2.17.2

Add $8$ and $1$ .

$\frac{9}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$

Step 6