Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$e^{y}]_{- 1}^{1} + \int_{- 1}^{1} - y^{2} d y + \int_{- 1}^{1} 2 d y$

Step 3

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

$e^{y}]_{- 1}^{1} - \int_{- 1}^{1} y^{2} d y + \int_{- 1}^{1} 2 d y$

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify the answer.

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Step 7.1

Combine $e^{y}]_{- 1}^{1}$ and $2 y]_{- 1}^{1}$ .

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $e^{y} + 2 y$ at $1$ and at $- 1$ .

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

Step 7.2.2

Evaluate $\frac{y^{3}}{3}$ at $1$ and at $- 1$ .

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify.

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

Step 7.2.3.2

Multiply $2$ by $1$ .

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

Step 7.2.3.3

Multiply $2$ by $- 1$ .

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

Step 7.2.3.4

One to any power is one.

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

Step 7.2.3.5

Raise $- 1$ to the power of $3$ .

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

Step 7.2.3.6

Move the negative in front of the fraction.

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

Step 7.2.3.7

Multiply $- 1$ by $- 1$ .

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

Step 7.2.3.8

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

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

Step 7.2.3.9

Combine the numerators over the common denominator.

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

Step 7.2.3.10

Add $1$ and $1$ .

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

Step 7.2.3.11

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

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

Step 7.2.3.12

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

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

Step 7.2.3.13

Combine the numerators over the common denominator.

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

Step 7.2.3.14

Simplify the numerator.

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Step 7.2.3.14.1

Multiply $2$ by $3$ .

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

Step 7.2.3.14.2

Subtract $2$ from $6$ .

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

Step 7.2.3.15

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

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

Step 7.2.3.16

Combine $- (e^{- 1} - 2)$ and $\frac{3}{3}$ .

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

Step 7.2.3.17

Combine the numerators over the common denominator.

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

Step 7.2.3.18

Multiply $3$ by $- 1$ .

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

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

Simplify the numerator.

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Step 8.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.1.1.2

Apply the distributive property.

$e + \frac{4 - 3 \frac{1}{e} - 3 \cdot - 2}{3}$

Step 8.1.1.3

Combine $- 3$ and $\frac{1}{e}$ .

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

Step 8.1.1.4

Multiply $- 3$ by $- 2$ .

$e + \frac{4 + \frac{- 3}{e} + 6}{3}$

Step 8.1.1.5

Move the negative in front of the fraction.

$e + \frac{4 - \frac{3}{e} + 6}{3}$

Step 8.1.1.6

Add $4$ and $6$ .

$e + \frac{10 - \frac{3}{e}}{3}$

Step 8.1.1.7

To write $10$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$e + \frac{\frac{10 e}{e} - \frac{3}{e}}{3}$

Step 8.1.1.8

Combine the numerators over the common denominator.

$e + \frac{\frac{10 e - 3}{e}}{3}$

Step 8.1.2

Multiply the numerator by the reciprocal of the denominator.

$e + \frac{10 e - 3}{e} \cdot \frac{1}{3}$

Step 8.1.3

Multiply $\frac{10 e - 3}{e}$ by $\frac{1}{3}$ .

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

Step 8.1.4

Move $3$ to the left of $e$ .

$e + \frac{10 e - 3}{3 e}$

Step 8.2

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

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

Step 8.3

Combine $e$ and $\frac{3 e}{3 e}$ .

$\frac{e (3 e)}{3 e} + \frac{10 e - 3}{3 e}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{e (3 e) + 10 e - 3}{3 e}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Move $3$ to the left of $e$ .

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

Step 8.5.2

Multiply $3 e e$ .

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Step 8.5.2.1

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

$\frac{3 (e^{1} e) + 10 e - 3}{3 e}$

Step 8.5.2.2

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

$\frac{3 (e^{1} e^{1}) + 10 e - 3}{3 e}$

Step 8.5.2.3

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

$\frac{3 e^{1 + 1} + 10 e - 3}{3 e}$

Step 8.5.2.4

Add $1$ and $1$ .

$\frac{3 e^{2} + 10 e - 3}{3 e}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3 e^{2} + 10 e - 3}{3 e}$

Decimal Form:

$5.68373572 \dots$

Step 10