Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x + 1$ and $d v = e^{- 2 x}$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

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

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

Step 5

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

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $- 2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.2

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

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

Step 6.3

Multiply $- 2$ by $1$ .

$u_{lower} = - 2$

Step 6.4

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

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

Step 6.5

Multiply $- 2$ by $2$ .

$u_{upper} = - 4$

Step 6.6

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

$u_{lower} = - 2$

$u_{upper} = - 4$

Step 6.7

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} + \frac{1}{2} \int_{- 2}^{- 4} e^{u} \frac{1}{- 2} d u$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} + \frac{1}{2} \int_{- 2}^{- 4} e^{u} (- \frac{1}{2}) d u$

Step 7.2

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} + \frac{1}{2} \int_{- 2}^{- 4} - \frac{e^{u}}{2} d u$

Step 8

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} + \frac{1}{2} (- \int_{- 2}^{- 4} \frac{e^{u}}{2} d u)$

Step 9

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} + \frac{1}{2} (- (\frac{1}{2} \int_{- 2}^{- 4} e^{u} d u))$

Step 10

Simplify.

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Step 10.1

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} - \frac{1}{2 \cdot 2} \int_{- 2}^{- 4} e^{u} d u$

Step 10.2

Multiply $2$ by $2$ .

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} - \frac{1}{4} \int_{- 2}^{- 4} e^{u} d u$

Step 11

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

$(x + 1) (- \frac{e^{- 2 x}}{2})]_{1}^{2} - \frac{1}{4} e^{u}]_{- 2}^{- 4}$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $(x + 1) (- \frac{e^{- 2 x}}{2})$ at $2$ and at $1$ .

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

Step 12.2

Evaluate $e^{u}$ at $- 4$ and at $- 2$ .

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

Step 12.3

Simplify.

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Step 12.3.1

Add $2$ and $1$ .

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

Step 12.3.2

Multiply $- 2$ by $2$ .

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

Step 12.3.3

Move $e^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12.3.4

Multiply $- 1$ by $3$ .

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

Step 12.3.5

Combine $- 3$ and $\frac{1}{2 e^{4}}$ .

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

Step 12.3.6

Move the negative in front of the fraction.

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

Step 12.3.7

Add $1$ and $1$ .

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

Step 12.3.8

Multiply $- 1$ by $2$ .

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

Step 12.3.9

Multiply $- 2$ by $1$ .

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

Step 12.3.10

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

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

Step 12.3.11

Multiply $- 1$ by $- 2$ .

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

Step 12.3.12

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

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

Step 12.3.13

Cancel the common factor of $2$ .

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Step 12.3.13.1

Cancel the common factor.

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

Step 12.3.13.2

Rewrite the expression.

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

Step 12.3.14

To write $- \frac{1}{4} (e^{- 4} - e^{- 2})$ as a fraction with a common denominator, multiply by $\frac{2 e^{4}}{2 e^{4}}$ .

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

Step 12.3.15

Combine $- \frac{1}{4} (e^{- 4} - e^{- 2})$ and $\frac{2 e^{4}}{2 e^{4}}$ .

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

Step 12.3.16

Combine the numerators over the common denominator.

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

Step 12.3.17

Multiply $2$ by $- 1$ .

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

Step 12.3.18

Combine $- 2$ and $\frac{1}{4}$ .

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

Step 12.3.19

Combine $e^{4}$ and $\frac{- 2}{4}$ .

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

Step 12.3.20

Move $- 2$ to the left of $e^{4}$ .

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

Step 12.3.21

Cancel the common factor of $- 2$ and $4$ .

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Step 12.3.21.1

Factor $2$ out of $- 2 e^{4}$ .

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

Step 12.3.21.2

Cancel the common factors.

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Step 12.3.21.2.1

Factor $2$ out of $4$ .

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

Step 12.3.21.2.2

Cancel the common factor.

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

Step 12.3.21.2.3

Rewrite the expression.

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

Step 12.3.22

Move the negative in front of the fraction.

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

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

Simplify the numerator.

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Step 13.1.1.1

Apply the distributive property.

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

Step 13.1.1.2

Multiply $- \frac{e^{4}}{2} e^{- 4}$ .

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Step 13.1.1.2.1

Combine $e^{- 4}$ and $\frac{e^{4}}{2}$ .

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

Step 13.1.1.2.2

Multiply $e^{- 4}$ by $e^{4}$ by adding the exponents.

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Step 13.1.1.2.2.1

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

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

Step 13.1.1.2.2.2

Add $- 4$ and $4$ .

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

Step 13.1.1.2.3

Simplify $e^{0}$ .

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

Step 13.1.1.3

Multiply $- \frac{e^{4}}{2} (- e^{- 2})$ .

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Step 13.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 13.1.1.3.2

Multiply $\frac{e^{4}}{2}$ by $1$ .

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

Step 13.1.1.3.3

Combine $\frac{e^{4}}{2}$ and $e^{- 2}$ .

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

Step 13.1.1.3.4

Multiply $e^{4}$ by $e^{- 2}$ by adding the exponents.

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Step 13.1.1.3.4.1

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

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

Step 13.1.1.3.4.2

Subtract $2$ from $4$ .

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

Step 13.1.1.4

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

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

Step 13.1.1.5

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

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

Step 13.1.1.6

Combine the numerators over the common denominator.

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

Step 13.1.1.7

Simplify the numerator.

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Step 13.1.1.7.1

Multiply $- 3$ by $2$ .

$\frac{1}{e^{2}} + \frac{\frac{- 6 - 1}{2} + \frac{e^{2}}{2}}{2 e^{4}}$

Step 13.1.1.7.2

Subtract $1$ from $- 6$ .

$\frac{1}{e^{2}} + \frac{\frac{- 7}{2} + \frac{e^{2}}{2}}{2 e^{4}}$

Step 13.1.1.8

Combine the numerators over the common denominator.

$\frac{1}{e^{2}} + \frac{\frac{- 7 + e^{2}}{2}}{2 e^{4}}$

Step 13.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{e^{2}} + \frac{- 7 + e^{2}}{2} \cdot \frac{1}{2 e^{4}}$

Step 13.1.3

Multiply $\frac{- 7 + e^{2}}{2} \cdot \frac{1}{2 e^{4}}$ .

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Step 13.1.3.1

Multiply $\frac{- 7 + e^{2}}{2}$ by $\frac{1}{2 e^{4}}$ .

$\frac{1}{e^{2}} + \frac{- 7 + e^{2}}{2 (2 e^{4})}$

Step 13.1.3.2

Multiply $2$ by $2$ .

$\frac{1}{e^{2}} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.2

To write $\frac{1}{e^{2}}$ as a fraction with a common denominator, multiply by $\frac{4 e^{2}}{4 e^{2}}$ .

$\frac{1}{e^{2}} \cdot \frac{4 e^{2}}{4 e^{2}} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.3

Write each expression with a common denominator of $4 e^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.3.1

Multiply $\frac{1}{e^{2}}$ by $\frac{4 e^{2}}{4 e^{2}}$ .

$\frac{4 e^{2}}{e^{2} (4 e^{2})} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.3.2

Multiply $e^{2}$ by $e^{2}$ by adding the exponents.

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Step 13.3.2.1

Move $e^{2}$ .

$\frac{4 e^{2}}{e^{2} e^{2} \cdot 4} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.3.2.2

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

$\frac{4 e^{2}}{e^{2 + 2} \cdot 4} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.3.2.3

Add $2$ and $2$ .

$\frac{4 e^{2}}{e^{4} \cdot 4} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.3.3

Reorder the factors of $e^{4} \cdot 4$ .

$\frac{4 e^{2}}{4 e^{4}} + \frac{- 7 + e^{2}}{4 e^{4}}$

Step 13.4

Combine the numerators over the common denominator.

$\frac{4 e^{2} - 7 + e^{2}}{4 e^{4}}$

Step 13.5

Add $4 e^{2}$ and $e^{2}$ .

$\frac{5 e^{2} - 7}{4 e^{4}}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{5 e^{2} - 7}{4 e^{4}}$

Decimal Form:

$0.13711673 \dots$

Step 15