Calculus Examples

$\int_{- 2}^{0} 4 x^{2} + 4 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{0} 4 x^{2} d x + \int_{- 2}^{0} 4 x d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{- 2}^{0} x^{2} d x + \int_{- 2}^{0} 4 x d x$

Step 3

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

$4 (\frac{1}{3} x^{3}]_{- 2}^{0}) + \int_{- 2}^{0} 4 x d x$

Step 4

Combine $\frac{1}{3}$ and $x^{3}$ .

$4 (\frac{x^{3}}{3}]_{- 2}^{0}) + \int_{- 2}^{0} 4 x d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 (\frac{x^{3}}{3}]_{- 2}^{0}) + 4 \int_{- 2}^{0} x d x$

Step 6

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

$4 (\frac{x^{3}}{3}]_{- 2}^{0}) + 4 (\frac{1}{2} x^{2}]_{- 2}^{0})$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$4 (\frac{x^{3}}{3}]_{- 2}^{0}) + 4 (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 7.2

Substitute and simplify.

Tap for more steps...

Step 7.2.1

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

$4 ((\frac{0^{3}}{3}) - \frac{{(- 2)}^{3}}{3}) + 4 (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 7.2.2

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

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

Step 7.2.3

Simplify.

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 7.2.3.2.1

Factor $3$ out of $0$ .

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

Step 7.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.2.2.1

Factor $3$ out of $3$ .

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

Step 7.2.3.2.2.2

Cancel the common factor.

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

Step 7.2.3.2.2.3

Rewrite the expression.

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

Step 7.2.3.2.2.4

Divide $0$ by $1$ .

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

Step 7.2.3.3

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

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

Step 7.2.3.4

Move the negative in front of the fraction.

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

Step 7.2.3.5

Multiply $- 1$ by $- 1$ .

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

Step 7.2.3.6

Multiply $\frac{8}{3}$ by $1$ .

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

Step 7.2.3.7

Add $0$ and $\frac{8}{3}$ .

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

Step 7.2.3.8

Combine $4$ and $\frac{8}{3}$ .

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

Step 7.2.3.9

Multiply $4$ by $8$ .

$\frac{32}{3} + 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.10

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

$\frac{32}{3} + 4 (\frac{0}{2} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.11

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 7.2.3.11.1

Factor $2$ out of $0$ .

$\frac{32}{3} + 4 (\frac{2 (0)}{2} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.11.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.11.2.1

Factor $2$ out of $2$ .

$\frac{32}{3} + 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.11.2.2

Cancel the common factor.

$\frac{32}{3} + 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.11.2.3

Rewrite the expression.

$\frac{32}{3} + 4 (\frac{0}{1} - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.11.2.4

Divide $0$ by $1$ .

$\frac{32}{3} + 4 (0 - \frac{{(- 2)}^{2}}{2})$

Step 7.2.3.12

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

$\frac{32}{3} + 4 (0 - \frac{4}{2})$

Step 7.2.3.13

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

Tap for more steps...

Step 7.2.3.13.1

Factor $2$ out of $4$ .

$\frac{32}{3} + 4 (0 - \frac{2 \cdot 2}{2})$

Step 7.2.3.13.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.13.2.1

Factor $2$ out of $2$ .

$\frac{32}{3} + 4 (0 - \frac{2 \cdot 2}{2 (1)})$

Step 7.2.3.13.2.2

Cancel the common factor.

$\frac{32}{3} + 4 (0 - \frac{2 \cdot 2}{2 \cdot 1})$

Step 7.2.3.13.2.3

Rewrite the expression.

$\frac{32}{3} + 4 (0 - \frac{2}{1})$

Step 7.2.3.13.2.4

Divide $2$ by $1$ .

$\frac{32}{3} + 4 (0 - 1 \cdot 2)$

Step 7.2.3.14

Multiply $- 1$ by $2$ .

$\frac{32}{3} + 4 (0 - 2)$

Step 7.2.3.15

Subtract $2$ from $0$ .

$\frac{32}{3} + 4 \cdot - 2$

Step 7.2.3.16

Multiply $4$ by $- 2$ .

$\frac{32}{3} - 8$

Step 7.2.3.17

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{32}{3} - 8 \cdot \frac{3}{3}$

Step 7.2.3.18

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

$\frac{32}{3} + \frac{- 8 \cdot 3}{3}$

Step 7.2.3.19

Combine the numerators over the common denominator.

$\frac{32 - 8 \cdot 3}{3}$

Step 7.2.3.20

Simplify the numerator.

Tap for more steps...

Step 7.2.3.20.1

Multiply $- 8$ by $3$ .

$\frac{32 - 24}{3}$

Step 7.2.3.20.2

Subtract $24$ from $32$ .

$\frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$

Step 9