Calculus Examples

$\int_{- 1}^{1} e^{x} - (x^{2} - 1) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 1}^{1} e^{x} d x + \int_{- 1}^{1} - (x^{2} - 1) d x$

Step 2

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

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

Step 3

Multiply $- (x^{2} - 1)$ .

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

Step 4

Multiply $- 1$ by $- 1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify the answer.

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Step 10.1

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

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

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.2.3

Simplify.

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Step 10.2.3.1

Simplify.

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

Step 10.2.3.2

One to any power is one.

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

Step 10.2.3.3

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

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

Step 10.2.3.4

Move the negative in front of the fraction.

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

Step 10.2.3.5

Multiply $- 1$ by $- 1$ .

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

Step 10.2.3.6

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

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

Step 10.2.3.7

Combine the numerators over the common denominator.

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

Step 10.2.3.8

Add $1$ and $1$ .

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

Step 10.2.3.9

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

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

Step 10.2.3.10

Combine the numerators over the common denominator.

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

Step 10.2.3.11

Subtract $2$ from $3$ .

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

Step 10.2.3.12

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

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

Step 10.2.3.13

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

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

Step 10.2.3.14

Combine the numerators over the common denominator.

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

Step 10.2.3.15

Multiply $3$ by $- 1$ .

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

Step 11

Simplify.

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Step 11.1

Simplify each term.

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Step 11.1.1

Simplify the numerator.

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Step 11.1.1.1

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

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

Step 11.1.1.2

Apply the distributive property.

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

Step 11.1.1.3

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

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

Step 11.1.1.4

Multiply $- 3$ by $- 1$ .

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

Step 11.1.1.5

Move the negative in front of the fraction.

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

Step 11.1.1.6

Add $1$ and $3$ .

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

Step 11.1.1.7

To write $4$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

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

Step 11.1.1.8

Combine the numerators over the common denominator.

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

Step 11.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.1.3

Multiply $\frac{4 e - 3}{e}$ by $\frac{1}{3}$ .

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

Step 11.1.4

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Simplify the numerator.

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Step 11.5.1

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

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

Step 11.5.2

Multiply $3 e e$ .

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Step 11.5.2.1

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

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

Step 11.5.2.2

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

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

Step 11.5.2.3

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

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

Step 11.5.2.4

Add $1$ and $1$ .

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.68373572 \dots$

Step 13