Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Let $u = y + 2$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = y + 2$ . Find $\frac{d u}{d y}$ .

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Step 4.1.1

Differentiate $y + 2$ .

$\frac{d}{d y} [y + 2]$

Step 4.1.2

By the Sum Rule, the derivative of $y + 2$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [2]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 4.1.3

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

$1 + \frac{d}{d y} [2]$

Step 4.1.4

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Substitute the lower limit in for $y$ in $u = y + 2$ .

$u_{lower} = - 1 + 2$

Step 4.3

Add $- 1$ and $2$ .

$u_{lower} = 1$

Step 4.4

Substitute the upper limit in for $y$ in $u = y + 2$ .

$u_{upper} = 0 + 2$

Step 4.5

Add $0$ and $2$ .

$u_{upper} = 2$

Step 4.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 4.7

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

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

Step 5

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

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

Raise $- 1$ to the power of $2$ .

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

Step 7.3.4

Multiply $- 1$ by $1$ .

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

Step 7.3.5

Subtract $\frac{1}{2}$ from $0$ .

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

Step 7.3.6

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

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

Step 7.3.7

Multiply $2$ by $2^{\frac{3}{2}}$ by adding the exponents.

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Step 7.3.7.1

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

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Step 7.3.7.1.1

Raise $2$ to the power of $1$ .

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

Step 7.3.7.1.2

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

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

Step 7.3.7.2

Write $1$ as a fraction with a common denominator.

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

Step 7.3.7.3

Combine the numerators over the common denominator.

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

Step 7.3.7.4

Add $2$ and $3$ .

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

Step 7.3.8

One to any power is one.

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

Step 7.3.9

Multiply $- 1$ by $1$ .

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

Step 7.3.10

Combine the numerators over the common denominator.

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

Step 7.3.11

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.3.12

To write $\frac{2^{\frac{5}{2}} - 2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.3.13

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.13.1

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

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

Step 7.3.13.2

Multiply $2$ by $3$ .

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

Step 7.3.13.3

Multiply $\frac{2^{\frac{5}{2}} - 2}{3}$ by $\frac{2}{2}$ .

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

Step 7.3.13.4

Multiply $3$ by $2$ .

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

Step 7.3.14

Combine the numerators over the common denominator.

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

Step 7.3.15

Move $2$ to the left of $2^{\frac{5}{2}} - 2$ .

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

Step 8

Simplify.

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Step 8.1

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- 1 (3) + 2 (2^{\frac{5}{2}} - 2)}{6}$

Step 8.2

Factor $- 1$ out of $2 (2^{\frac{5}{2}} - 2)$ .

$\frac{- 1 (3) - (- 2 (2^{\frac{5}{2}} - 2))}{6}$

Step 8.3

Factor $- 1$ out of $- 1 (3) - (- 2 (2^{\frac{5}{2}} - 2))$ .

$\frac{- 1 (3 - 2 (2^{\frac{5}{2}} - 2))}{6}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{3 - 2 (2^{\frac{5}{2}} - 2)}{6}$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $- 2 \cdot 2^{\frac{5}{2}}$ .

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Step 9.2.1

Factor out negative.

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

Step 9.2.2

Raise $2$ to the power of $1$ .

$- \frac{3 - (2^{1} \cdot 2^{\frac{5}{2}}) - 2 \cdot - 2}{6}$

Step 9.2.3

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

$- \frac{3 - 2^{1 + \frac{5}{2}} - 2 \cdot - 2}{6}$

Step 9.2.4

Write $1$ as a fraction with a common denominator.

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

Step 9.2.5

Combine the numerators over the common denominator.

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

Step 9.2.6

Add $2$ and $5$ .

$- \frac{3 - 2^{\frac{7}{2}} - 2 \cdot - 2}{6}$

Step 9.3

Multiply $- 2$ by $- 2$ .

$- \frac{3 - 2^{\frac{7}{2}} + 4}{6}$

Step 9.4

Add $3$ and $4$ .

$- \frac{7 - 2^{\frac{7}{2}}}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{7 - 2^{\frac{7}{2}}}{6}$

Decimal Form:

$0.71895141 \dots$

Step 11