Calculus Examples

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

Step 1

Remove parentheses.

$\int_{- 2}^{3} x^{2} - 3 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3} x^{3}]_{- 2}^{3}$ and $- 3 x]_{- 2}^{3}$ .

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$\frac{1}{3} \cdot 27 - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.3

Cancel the common factor of $27$ and $3$ .

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Step 5.2.2.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.3.2

Cancel the common factors.

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Step 5.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.3.2.3

Rewrite the expression.

$\frac{9}{1} - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.3.2.4

Divide $9$ by $1$ .

$9 - 3 \cdot 3 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.4

Multiply $- 3$ by $3$ .

$9 - 9 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.5

Subtract $9$ from $9$ .

$0 - (\frac{1}{3} {(- 2)}^{3} - 3 \cdot - 2)$

Step 5.2.2.6

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

$0 - (\frac{1}{3} \cdot - 8 - 3 \cdot - 2)$

Step 5.2.2.7

Combine $\frac{1}{3}$ and $- 8$ .

$0 - (\frac{- 8}{3} - 3 \cdot - 2)$

Step 5.2.2.8

Move the negative in front of the fraction.

$0 - (- \frac{8}{3} - 3 \cdot - 2)$

Step 5.2.2.9

Multiply $- 3$ by $- 2$ .

$0 - (- \frac{8}{3} + 6)$

Step 5.2.2.10

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 - (- \frac{8}{3} + 6 \cdot \frac{3}{3})$

Step 5.2.2.11

Combine $6$ and $\frac{3}{3}$ .

$0 - (- \frac{8}{3} + \frac{6 \cdot 3}{3})$

Step 5.2.2.12

Combine the numerators over the common denominator.

$0 - \frac{- 8 + 6 \cdot 3}{3}$

Step 5.2.2.13

Simplify the numerator.

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Step 5.2.2.13.1

Multiply $6$ by $3$ .

$0 - \frac{- 8 + 18}{3}$

Step 5.2.2.13.2

Add $- 8$ and $18$ .

$0 - \frac{10}{3}$

Step 5.2.2.14

Subtract $\frac{10}{3}$ from $0$ .

$- \frac{10}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{10}{3}$

Decimal Form:

$- 3.\overline{3}$

Mixed Number Form:

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

Step 7