Calculus Examples

$\int_{- 4}^{0} (\frac{y}{4} + \sqrt{y + 5}) d y$

Step 1

Remove parentheses.

$\int_{- 4}^{0} \frac{y}{4} + \sqrt{y + 5} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 4}^{0} \frac{y}{4} d y + \int_{- 4}^{0} \sqrt{y + 5} d y$

Step 3

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{- 4}^{0} y d y + \int_{- 4}^{0} \sqrt{y + 5} d y$

Step 4

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

$\frac{1}{4} \frac{1}{2} y^{2}]_{- 4}^{0} + \int_{- 4}^{0} \sqrt{y + 5} d y$

Step 5

Let $u = y + 5$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = y + 5$ . Find $\frac{d u}{d y}$ .

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Step 5.1.1

Differentiate $y + 5$ .

$\frac{d}{d y} [y + 5]$

Step 5.1.2

By the Sum Rule, the derivative of $y + 5$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [5]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [5]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [5]$

Step 5.1.4

Since $5$ is constant with respect to $y$ , the derivative of $5$ with respect to $y$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Substitute the lower limit in for $y$ in $u = y + 5$ .

$u_{lower} = - 4 + 5$

Step 5.3

Add $- 4$ and $5$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for $y$ in $u = y + 5$ .

$u_{upper} = 0 + 5$

Step 5.5

Add $0$ and $5$ .

$u_{upper} = 5$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 5$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{1}{4} \frac{1}{2} y^{2}]_{- 4}^{0} + \int_{1}^{5} \sqrt{u} d u$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\frac{1}{4} \frac{1}{2} y^{2}]_{- 4}^{0} + \int_{1}^{5} u^{\frac{1}{2}} d u$

Step 7

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

$\frac{1}{4} \frac{1}{2} y^{2}]_{- 4}^{0} + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{2} y^{2}$ at $0$ and at $- 4$ .

$\frac{1}{4} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 4)}^{2}) + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}$

Step 8.2

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $5$ and at $1$ .

$\frac{1}{4} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 4)}^{2}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{1}{4} (\frac{1}{2} \cdot 0 - \frac{1}{2} {(- 4)}^{2}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.2

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

$\frac{1}{4} (0 - \frac{1}{2} {(- 4)}^{2}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.3

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

$\frac{1}{4} (0 - \frac{1}{2} \cdot 16) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.4

Multiply $16$ by $- 1$ .

$\frac{1}{4} (0 - 16 (\frac{1}{2})) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.5

Combine $- 16$ and $\frac{1}{2}$ .

$\frac{1}{4} (0 + \frac{- 16}{2}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6

Cancel the common factor of $- 16$ and $2$ .

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Step 8.3.6.1

Factor $2$ out of $- 16$ .

$\frac{1}{4} (0 + \frac{2 \cdot - 8}{2}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2

Cancel the common factors.

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Step 8.3.6.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} (0 + \frac{2 \cdot - 8}{2 (1)}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.2

Cancel the common factor.

$\frac{1}{4} (0 + \frac{2 \cdot - 8}{2 \cdot 1}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.3

Rewrite the expression.

$\frac{1}{4} (0 + \frac{- 8}{1}) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.4

Divide $- 8$ by $1$ .

$\frac{1}{4} (0 - 8) + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.7

Subtract $8$ from $0$ .

$\frac{1}{4} \cdot - 8 + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.8

Combine $\frac{1}{4}$ and $- 8$ .

$\frac{- 8}{4} + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9

Cancel the common factor of $- 8$ and $4$ .

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Step 8.3.9.1

Factor $4$ out of $- 8$ .

$\frac{4 \cdot - 2}{4} + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2

Cancel the common factors.

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Step 8.3.9.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 2}{4 (1)} + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.2

Cancel the common factor.

$\frac{4 \cdot - 2}{4 \cdot 1} + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.3

Rewrite the expression.

$\frac{- 2}{1} + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.4

Divide $- 2$ by $1$ .

$- 2 + (\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10

Combine $\frac{2}{3}$ and $5^{\frac{3}{2}}$ .

$- 2 + \frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.11

One to any power is one.

$- 2 + \frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1$

Step 8.3.12

Multiply $- 1$ by $1$ .

$- 2 + \frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3}$

Step 8.3.13

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

$- 2 \cdot \frac{3}{3} - \frac{2}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8.3.14

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

$\frac{- 2 \cdot 3}{3} - \frac{2}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8.3.15

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 3 - 2}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8.3.16

Simplify the numerator.

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Step 8.3.16.1

Multiply $- 2$ by $3$ .

$\frac{- 6 - 2}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8.3.16.2

Subtract $2$ from $- 6$ .

$\frac{- 8}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8.3.17

Move the negative in front of the fraction.

$- \frac{8}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{3} + \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Decimal Form:

$4.78689325 \dots$

Step 10