Calculus Examples

$\int_{- 2}^{0} (x^{2} + x) d x$

Step 1

Remove parentheses.

$\int_{- 2}^{0} x^{2} + x d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{0} x^{2} d x + \int_{- 2}^{0} x d x$

Step 3

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

$\frac{1}{3} x^{3}]_{- 2}^{0} + \int_{- 2}^{0} x d x$

Step 4

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

$\frac{1}{3} x^{3}]_{- 2}^{0} + \frac{1}{2} x^{2}]_{- 2}^{0}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Combine $\frac{1}{3} x^{3}]_{- 2}^{0}$ and $\frac{1}{2} x^{2}]_{- 2}^{0}$ .

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{- 2}^{0}$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}$ at $0$ and at $- 2$ .

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

Step 5.2.2

Simplify.

Tap for more steps...

Step 5.2.2.1

Raising $0$ to any positive power yields $0$ .

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

Step 5.2.2.2

Multiply $\frac{1}{3}$ by $0$ .

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

Step 5.2.2.3

Raising $0$ to any positive power yields $0$ .

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

Step 5.2.2.4

Multiply $\frac{1}{2}$ by $0$ .

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

Step 5.2.2.5

Add $0$ and $0$ .

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

Step 5.2.2.6

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

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

Step 5.2.2.7

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

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

Step 5.2.2.8

Move the negative in front of the fraction.

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

Step 5.2.2.9

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

$0 - (- \frac{8}{3} + \frac{1}{2} \cdot 4)$

Step 5.2.2.10

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

$0 - (- \frac{8}{3} + \frac{4}{2})$

Step 5.2.2.11

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

Tap for more steps...

Step 5.2.2.11.1

Factor $2$ out of $4$ .

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

Step 5.2.2.11.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.11.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.11.2.2

Cancel the common factor.

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

Step 5.2.2.11.2.3

Rewrite the expression.

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

Step 5.2.2.11.2.4

Divide $2$ by $1$ .

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

Step 5.2.2.12

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.2.13

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

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

Step 5.2.2.14

Combine the numerators over the common denominator.

$0 - \frac{- 8 + 2 \cdot 3}{3}$

Step 5.2.2.15

Simplify the numerator.

Tap for more steps...

Step 5.2.2.15.1

Multiply $2$ by $3$ .

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

Step 5.2.2.15.2

Add $- 8$ and $6$ .

$0 - \frac{- 2}{3}$

Step 5.2.2.16

Move the negative in front of the fraction.

$0 - - \frac{2}{3}$

Step 5.2.2.17

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{2}{3})$

Step 5.2.2.18

Multiply $\frac{2}{3}$ by $1$ .

$0 + \frac{2}{3}$

Step 5.2.2.19

Add $0$ and $\frac{2}{3}$ .

$\frac{2}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Step 7