Calculus Examples

$\int_{0}^{1} (e^{x} - e^{- 2 x}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} e^{x} - e^{- 2 x} d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$e^{x}]_{0}^{1} + \int_{0}^{1} - e^{- 2 x} d x$

Step 4

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

$e^{x}]_{0}^{1} - \int_{0}^{1} e^{- 2 x} d x$

Step 5

Let $u = - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 5.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

Step 5.1.3

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

$- 2 \cdot 1$

Step 5.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 5.2

Substitute the lower limit in for $x$ in $u = - 2 x$ .

$u_{lower} = - 2 \cdot 0$

Step 5.3

Multiply $- 2$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = - 2 x$ .

$u_{upper} = - 2 \cdot 1$

Step 5.5

Multiply $- 2$ by $1$ .

$u_{upper} = - 2$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = - 2$

Step 5.7

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

$e^{x}]_{0}^{1} - \int_{0}^{- 2} e^{u} \frac{1}{- 2} d u$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

$e^{x}]_{0}^{1} - \int_{0}^{- 2} e^{u} (- \frac{1}{2}) d u$

Step 6.2

Combine $e^{u}$ and $\frac{1}{2}$ .

$e^{x}]_{0}^{1} - \int_{0}^{- 2} - \frac{e^{u}}{2} d u$

Step 7

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

$e^{x}]_{0}^{1} - - \int_{0}^{- 2} \frac{e^{u}}{2} d u$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int_{0}^{- 2} \frac{e^{u}}{2} d u$ by $1$ .

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

Step 9

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 10

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

$e^{x}]_{0}^{1} + \frac{1}{2} e^{u}]_{0}^{- 2}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $e^{x}$ at $1$ and at $0$ .

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

Step 11.2

Evaluate $e^{u}$ at $- 2$ and at $0$ .

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

Step 11.3

Simplify.

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Step 11.3.1

Simplify.

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

Step 11.3.2

Anything raised to $0$ is $1$ .

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

Step 11.3.3

Multiply $- 1$ by $1$ .

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

Step 11.3.4

Anything raised to $0$ is $1$ .

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

Step 11.3.5

Multiply $- 1$ by $1$ .

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

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Apply the distributive property.

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

Step 12.1.2

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

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

Step 12.1.3

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

$e - 1 + \frac{e^{- 2}}{2} + \frac{- 1}{2}$

Step 12.1.4

Simplify each term.

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Step 12.1.4.1

Move $e^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e - 1 + \frac{1}{2 e^{2}} + \frac{- 1}{2}$

Step 12.1.4.2

Move the negative in front of the fraction.

$e - 1 + \frac{1}{2 e^{2}} - \frac{1}{2}$

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Simplify the numerator.

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Step 12.5.1

Multiply $- 1$ by $2$ .

$e + \frac{1}{2 e^{2}} + \frac{- 2 - 1}{2}$

Step 12.5.2

Subtract $1$ from $- 2$ .

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

Step 12.6

Move the negative in front of the fraction.

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.28594947 \dots$

Step 14