Calculus Examples

$\int_{0}^{2} (y - 1) (2 y + 1) d y$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Apply the distributive property.

$\int_{0}^{2} y (2 y + 1) - 1 (2 y + 1) d y$

Step 1.2

Apply the distributive property.

$\int_{0}^{2} y (2 y) + y \cdot 1 - 1 (2 y + 1) d y$

Step 1.3

Apply the distributive property.

$\int_{0}^{2} y (2 y) + y \cdot 1 - 1 (2 y) - 1 \cdot 1 d y$

Step 1.4

Reorder $y$ and $2$ .

$\int_{0}^{2} 2 \cdot y \cdot y + y \cdot 1 - 1 (2 y) - 1 \cdot 1 d y$

Step 1.5

Reorder $y$ and $1$ .

$\int_{0}^{2} 2 \cdot y \cdot y + 1 \cdot y - 1 (2 y) - 1 \cdot 1 d y$

Step 1.6

Raise $y$ to the power of $1$ .

$\int_{0}^{2} 2 (y^{1} y) + 1 y - 1 \cdot 2 y - 1 \cdot 1 d y$

Step 1.7

Raise $y$ to the power of $1$ .

$\int_{0}^{2} 2 (y^{1} y^{1}) + 1 y - 1 \cdot 2 y - 1 \cdot 1 d y$

Step 1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{2} 2 y^{1 + 1} + 1 y - 1 \cdot 2 y - 1 \cdot 1 d y$

Step 1.9

Add $1$ and $1$ .

$\int_{0}^{2} 2 y^{2} + 1 y - 1 \cdot 2 y - 1 \cdot 1 d y$

Step 1.10

Multiply $y$ by $1$ .

$\int_{0}^{2} 2 y^{2} + y - 1 \cdot 2 y - 1 \cdot 1 d y$

Step 1.11

Multiply $- 1$ by $2$ .

$\int_{0}^{2} 2 y^{2} + y - 2 y - 1 \cdot 1 d y$

Step 1.12

Multiply $- 1$ by $1$ .

$\int_{0}^{2} 2 y^{2} + y - 2 y - 1 d y$

Step 1.13

Subtract $2 y$ from $y$ .

$\int_{0}^{2} 2 y^{2} - y - 1 d y$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2} 2 y^{2} d y + \int_{0}^{2} - y d y + \int_{0}^{2} - 1 d y$

Step 3

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int_{0}^{2} y^{2} d y + \int_{0}^{2} - y d y + \int_{0}^{2} - 1 d y$

Step 4

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

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

Step 5

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

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

Step 6

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Substitute and simplify.

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Evaluate $- y$ at $2$ and at $0$ .

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

Step 10.4

Simplify.

Tap for more steps...

Step 10.4.1

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

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

Step 10.4.2

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

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

Step 10.4.3

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

Tap for more steps...

Step 10.4.3.1

Factor $3$ out of $0$ .

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

Step 10.4.3.2

Cancel the common factors.

Tap for more steps...

Step 10.4.3.2.1

Factor $3$ out of $3$ .

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

Step 10.4.3.2.2

Cancel the common factor.

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

Step 10.4.3.2.3

Rewrite the expression.

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

Step 10.4.3.2.4

Divide $0$ by $1$ .

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

Step 10.4.4

Multiply $- 1$ by $0$ .

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

Step 10.4.5

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

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

Step 10.4.6

Combine $2$ and $\frac{8}{3}$ .

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

Step 10.4.7

Multiply $2$ by $8$ .

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

Step 10.4.8

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

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

Step 10.4.9

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

Tap for more steps...

Step 10.4.9.1

Factor $2$ out of $4$ .

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

Step 10.4.9.2

Cancel the common factors.

Tap for more steps...

Step 10.4.9.2.1

Factor $2$ out of $2$ .

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

Step 10.4.9.2.2

Cancel the common factor.

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

Step 10.4.9.2.3

Rewrite the expression.

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

Step 10.4.9.2.4

Divide $2$ by $1$ .

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

Step 10.4.10

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

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

Step 10.4.11

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

Tap for more steps...

Step 10.4.11.1

Factor $2$ out of $0$ .

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

Step 10.4.11.2

Cancel the common factors.

Tap for more steps...

Step 10.4.11.2.1

Factor $2$ out of $2$ .

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

Step 10.4.11.2.2

Cancel the common factor.

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

Step 10.4.11.2.3

Rewrite the expression.

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

Step 10.4.11.2.4

Divide $0$ by $1$ .

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

Step 10.4.12

Multiply $- 1$ by $0$ .

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

Step 10.4.13

Add $2$ and $0$ .

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

Step 10.4.14

Multiply $- 1$ by $2$ .

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

Step 10.4.15

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

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

Step 10.4.16

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

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

Step 10.4.17

Combine the numerators over the common denominator.

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

Step 10.4.18

Simplify the numerator.

Tap for more steps...

Step 10.4.18.1

Multiply $- 2$ by $3$ .

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

Step 10.4.18.2

Subtract $6$ from $16$ .

$\frac{10}{3} + (- 1 \cdot 2) + 0$

Step 10.4.19

Multiply $- 1$ by $2$ .

$\frac{10}{3} - 2 + 0$

Step 10.4.20

Add $- 2$ and $0$ .

$\frac{10}{3} - 2$

Step 10.4.21

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

$\frac{10}{3} - 2 \cdot \frac{3}{3}$

Step 10.4.22

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

$\frac{10}{3} + \frac{- 2 \cdot 3}{3}$

Step 10.4.23

Combine the numerators over the common denominator.

$\frac{10 - 2 \cdot 3}{3}$

Step 10.4.24

Simplify the numerator.

Tap for more steps...

Step 10.4.24.1

Multiply $- 2$ by $3$ .

$\frac{10 - 6}{3}$

Step 10.4.24.2

Subtract $6$ from $10$ .

$\frac{4}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$

Step 12