Calculus Examples

$\int_{0}^{2} (8 z + 4 z^{3} - 6 z^{2}) d z$

Step 1

Remove parentheses.

$\int_{0}^{2} 8 z + 4 z^{3} - 6 z^{2} d z$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2} 8 z d z + \int_{0}^{2} 4 z^{3} d z + \int_{0}^{2} - 6 z^{2} d z$

Step 3

Since $8$ is constant with respect to $z$ , move $8$ out of the integral.

$8 \int_{0}^{2} z d z + \int_{0}^{2} 4 z^{3} d z + \int_{0}^{2} - 6 z^{2} d z$

Step 4

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

$8 (\frac{1}{2} z^{2}]_{0}^{2}) + \int_{0}^{2} 4 z^{3} d z + \int_{0}^{2} - 6 z^{2} d z$

Step 5

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

$8 (\frac{z^{2}}{2}]_{0}^{2}) + \int_{0}^{2} 4 z^{3} d z + \int_{0}^{2} - 6 z^{2} d z$

Step 6

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

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 \int_{0}^{2} z^{3} d z + \int_{0}^{2} - 6 z^{2} d z$

Step 7

By the Power Rule, the integral of $z^{3}$ with respect to $z$ is $\frac{1}{4} z^{4}$ .

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 (\frac{1}{4} z^{4}]_{0}^{2}) + \int_{0}^{2} - 6 z^{2} d z$

Step 8

Combine $\frac{1}{4}$ and $z^{4}$ .

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 (\frac{z^{4}}{4}]_{0}^{2}) + \int_{0}^{2} - 6 z^{2} d z$

Step 9

Since $- 6$ is constant with respect to $z$ , move $- 6$ out of the integral.

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 (\frac{z^{4}}{4}]_{0}^{2}) - 6 \int_{0}^{2} z^{2} d z$

Step 10

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

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 (\frac{z^{4}}{4}]_{0}^{2}) - 6 (\frac{1}{3} z^{3}]_{0}^{2})$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

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

$8 (\frac{z^{2}}{2}]_{0}^{2}) + 4 (\frac{z^{4}}{4}]_{0}^{2}) - 6 (\frac{z^{3}}{3}]_{0}^{2})$

Step 11.2

Substitute and simplify.

Tap for more steps...

Step 11.2.1

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

$8 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}) + 4 (\frac{z^{4}}{4}]_{0}^{2}) - 6 (\frac{z^{3}}{3}]_{0}^{2})$

Step 11.2.2

Evaluate $\frac{z^{4}}{4}$ at $2$ and at $0$ .

$8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 (\frac{z^{3}}{3}]_{0}^{2})$

Step 11.2.3

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

$8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4

Simplify.

Tap for more steps...

Step 11.2.4.1

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

$8 (\frac{4}{2} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.2

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

Tap for more steps...

Step 11.2.4.2.1

Factor $2$ out of $4$ .

$8 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.2.2.1

Factor $2$ out of $2$ .

$8 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.2.2.2

Cancel the common factor.

$8 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.2.2.3

Rewrite the expression.

$8 (\frac{2}{1} - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.2.2.4

Divide $2$ by $1$ .

$8 (2 - \frac{0^{2}}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.3

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

$8 (2 - \frac{0}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.4

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

Tap for more steps...

Step 11.2.4.4.1

Factor $2$ out of $0$ .

$8 (2 - \frac{2 (0)}{2}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.4.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.4.2.1

Factor $2$ out of $2$ .

$8 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.4.2.2

Cancel the common factor.

$8 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.4.2.3

Rewrite the expression.

$8 (2 - \frac{0}{1}) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.4.2.4

Divide $0$ by $1$ .

$8 (2 - 0) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.5

Multiply $- 1$ by $0$ .

$8 (2 + 0) + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.6

Add $2$ and $0$ .

$8 \cdot 2 + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.7

Multiply $8$ by $2$ .

$16 + 4 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.8

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

$16 + 4 (\frac{16}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.9

Cancel the common factor of $16$ and $4$ .

Tap for more steps...

Step 11.2.4.9.1

Factor $4$ out of $16$ .

$16 + 4 (\frac{4 \cdot 4}{4} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.9.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.9.2.1

Factor $4$ out of $4$ .

$16 + 4 (\frac{4 \cdot 4}{4 (1)} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.9.2.2

Cancel the common factor.

$16 + 4 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.9.2.3

Rewrite the expression.

$16 + 4 (\frac{4}{1} - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.9.2.4

Divide $4$ by $1$ .

$16 + 4 (4 - \frac{0^{4}}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.10

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

$16 + 4 (4 - \frac{0}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.11

Cancel the common factor of $0$ and $4$ .

Tap for more steps...

Step 11.2.4.11.1

Factor $4$ out of $0$ .

$16 + 4 (4 - \frac{4 (0)}{4}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.11.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.11.2.1

Factor $4$ out of $4$ .

$16 + 4 (4 - \frac{4 \cdot 0}{4 \cdot 1}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.11.2.2

Cancel the common factor.

$16 + 4 (4 - \frac{4 \cdot 0}{4 \cdot 1}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.11.2.3

Rewrite the expression.

$16 + 4 (4 - \frac{0}{1}) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.11.2.4

Divide $0$ by $1$ .

$16 + 4 (4 - 0) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.12

Multiply $- 1$ by $0$ .

$16 + 4 (4 + 0) - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.13

Add $4$ and $0$ .

$16 + 4 \cdot 4 - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.14

Multiply $4$ by $4$ .

$16 + 16 - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.15

Add $16$ and $16$ .

$32 - 6 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 11.2.4.16

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

$32 - 6 (\frac{8}{3} - \frac{0^{3}}{3})$

Step 11.2.4.17

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

$32 - 6 (\frac{8}{3} - \frac{0}{3})$

Step 11.2.4.18

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

Tap for more steps...

Step 11.2.4.18.1

Factor $3$ out of $0$ .

$32 - 6 (\frac{8}{3} - \frac{3 (0)}{3})$

Step 11.2.4.18.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.18.2.1

Factor $3$ out of $3$ .

$32 - 6 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 11.2.4.18.2.2

Cancel the common factor.

$32 - 6 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 11.2.4.18.2.3

Rewrite the expression.

$32 - 6 (\frac{8}{3} - \frac{0}{1})$

Step 11.2.4.18.2.4

Divide $0$ by $1$ .

$32 - 6 (\frac{8}{3} - 0)$

Step 11.2.4.19

Multiply $- 1$ by $0$ .

$32 - 6 (\frac{8}{3} + 0)$

Step 11.2.4.20

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

$32 - 6 (\frac{8}{3})$

Step 11.2.4.21

Combine $- 6$ and $\frac{8}{3}$ .

$32 + \frac{- 6 \cdot 8}{3}$

Step 11.2.4.22

Multiply $- 6$ by $8$ .

$32 + \frac{- 48}{3}$

Step 11.2.4.23

Cancel the common factor of $- 48$ and $3$ .

Tap for more steps...

Step 11.2.4.23.1

Factor $3$ out of $- 48$ .

$32 + \frac{3 \cdot - 16}{3}$

Step 11.2.4.23.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.23.2.1

Factor $3$ out of $3$ .

$32 + \frac{3 \cdot - 16}{3 (1)}$

Step 11.2.4.23.2.2

Cancel the common factor.

$32 + \frac{3 \cdot - 16}{3 \cdot 1}$

Step 11.2.4.23.2.3

Rewrite the expression.

$32 + \frac{- 16}{1}$

Step 11.2.4.23.2.4

Divide $- 16$ by $1$ .

$32 - 16$

Step 11.2.4.24

Subtract $16$ from $32$ .

$16$

Step 12