Calculus Examples

$\int_{- 4}^{- 1} x^{2} e^{x} d x$

Step 1

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

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

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 3

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

$x^{2} e^{x}]_{- 4}^{- 1} - 2 (x e^{x}]_{- 4}^{- 1} - (e^{x}]_{- 4}^{- 1}))$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $x^{2} e^{x}$ at $- 1$ and at $- 4$ .

$({(- 1)}^{2} e^{- 1}) - {(- 4)}^{2} e^{- 4} - 2 (x e^{x}]_{- 4}^{- 1} - (e^{x}]_{- 4}^{- 1}))$

Step 6.2

Evaluate $x e^{x}$ at $- 1$ and at $- 4$ .

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

Step 6.3

Evaluate $e^{x}$ at $- 1$ and at $- 4$ .

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

Step 6.4

Simplify.

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Step 6.4.1

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

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

Step 6.4.2

Multiply $e^{- 1}$ by $1$ .

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

Step 6.4.3

Raise $- 4$ to the power of $2$ .

$e^{- 1} - 1 \cdot 16 e^{- 4} - 2 ((- e^{- 1}) + 4 e^{- 4} - ((e^{- 1}) - e^{- 4}))$

Step 6.4.4

Multiply $- 1$ by $16$ .

$e^{- 1} - 16 e^{- 4} - 2 (- e^{- 1} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$\frac{1}{e} - 16 e^{- 4} - 2 (- e^{- 1} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7.1.2

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

$\frac{1}{e} - 16 \frac{1}{e^{4}} - 2 (- e^{- 1} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7.1.3

Combine $- 16$ and $\frac{1}{e^{4}}$ .

$\frac{1}{e} + \frac{- 16}{e^{4}} - 2 (- e^{- 1} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7.1.4

Move the negative in front of the fraction.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- e^{- 1} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7.1.5

Simplify each term.

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Step 7.1.5.1

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + 4 e^{- 4} - (e^{- 1} - e^{- 4}))$

Step 7.1.5.2

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + 4 \frac{1}{e^{4}} - (e^{- 1} - e^{- 4}))$

Step 7.1.5.3

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + \frac{4}{e^{4}} - (e^{- 1} - e^{- 4}))$

Step 7.1.5.4

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + \frac{4}{e^{4}} - (\frac{1}{e} - e^{- 4}))$

Step 7.1.5.5

Apply the distributive property.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + \frac{4}{e^{4}} - \frac{1}{e} - - e^{- 4})$

Step 7.1.5.6

Multiply $- - e^{- 4}$ .

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Step 7.1.5.6.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + \frac{4}{e^{4}} - \frac{1}{e} + 1 e^{- 4})$

Step 7.1.5.6.2

Multiply $e^{- 4}$ by $1$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (- \frac{1}{e} + \frac{4}{e^{4}} - \frac{1}{e} + e^{- 4})$

Step 7.1.6

Combine the numerators over the common denominator.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (e^{- 4} + \frac{- 1 - 1}{e} + \frac{4}{e^{4}})$

Step 7.1.7

Subtract $1$ from $- 1$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (e^{- 4} + \frac{- 2}{e} + \frac{4}{e^{4}})$

Step 7.1.8

Move the negative in front of the fraction.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 (e^{- 4} - \frac{2}{e} + \frac{4}{e^{4}})$

Step 7.1.9

Apply the distributive property.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} - 2 (- \frac{2}{e}) - 2 \frac{4}{e^{4}}$

Step 7.1.10

Simplify.

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Step 7.1.10.1

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

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Step 7.1.10.1.1

Multiply $- 1$ by $- 2$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + 2 \frac{2}{e} - 2 \frac{4}{e^{4}}$

Step 7.1.10.1.2

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + \frac{2 \cdot 2}{e} - 2 \frac{4}{e^{4}}$

Step 7.1.10.1.3

Multiply $2$ by $2$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + \frac{4}{e} - 2 \frac{4}{e^{4}}$

Step 7.1.10.2

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

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Step 7.1.10.2.1

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

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + \frac{4}{e} + \frac{- 2 \cdot 4}{e^{4}}$

Step 7.1.10.2.2

Multiply $- 2$ by $4$ .

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + \frac{4}{e} + \frac{- 8}{e^{4}}$

Step 7.1.11

Move the negative in front of the fraction.

$\frac{1}{e} - \frac{16}{e^{4}} - 2 e^{- 4} + \frac{4}{e} - \frac{8}{e^{4}}$

Step 7.2

Combine the numerators over the common denominator.

$- 2 e^{- 4} + \frac{1 + 4}{e} + \frac{- 16 - 8}{e^{4}}$

Step 7.3

Add $1$ and $4$ .

$- 2 e^{- 4} + \frac{5}{e} + \frac{- 16 - 8}{e^{4}}$

Step 7.4

Subtract $8$ from $- 16$ .

$- 2 e^{- 4} + \frac{5}{e} + \frac{- 24}{e^{4}}$

Step 7.5

Simplify each term.

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Step 7.5.1

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

$- 2 \frac{1}{e^{4}} + \frac{5}{e} + \frac{- 24}{e^{4}}$

Step 7.5.2

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

$\frac{- 2}{e^{4}} + \frac{5}{e} + \frac{- 24}{e^{4}}$

Step 7.5.3

Move the negative in front of the fraction.

$- \frac{2}{e^{4}} + \frac{5}{e} + \frac{- 24}{e^{4}}$

Step 7.5.4

Move the negative in front of the fraction.

$- \frac{2}{e^{4}} + \frac{5}{e} - \frac{24}{e^{4}}$

Step 7.6

Combine the numerators over the common denominator.

$\frac{- 2 - 24}{e^{4}} + \frac{5}{e}$

Step 7.7

Subtract $24$ from $- 2$ .

$\frac{- 26}{e^{4}} + \frac{5}{e}$

Step 7.8

Move the negative in front of the fraction.

$- \frac{26}{e^{4}} + \frac{5}{e}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{26}{e^{4}} + \frac{5}{e}$

Decimal Form:

$1.36319059 \dots$

Step 9