Calculus Examples

$\int_{- 3}^{4} (2 e^{- 3 x} - 3 e^{x}) d x$

Step 1

Remove parentheses.

$\int_{- 3}^{4} 2 e^{- 3 x} - 3 e^{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 3}^{4} 2 e^{- 3 x} d x + \int_{- 3}^{4} - 3 e^{x} d x$

Step 3

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

$2 \int_{- 3}^{4} e^{- 3 x} d x + \int_{- 3}^{4} - 3 e^{x} d x$

Step 4

Let $u = - 3 x$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 4.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 4.1.3

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

$- 3 \cdot 1$

Step 4.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 4.2

Substitute the lower limit in for $x$ in $u = - 3 x$ .

$u_{lower} = - 3 \cdot - 3$

Step 4.3

Multiply $- 3$ by $- 3$ .

$u_{lower} = 9$

Step 4.4

Substitute the upper limit in for $x$ in $u = - 3 x$ .

$u_{upper} = - 3 \cdot 4$

Step 4.5

Multiply $- 3$ by $4$ .

$u_{upper} = - 12$

Step 4.6

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

$u_{lower} = 9$

$u_{upper} = - 12$

Step 4.7

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

$2 \int_{9}^{- 12} e^{u} \frac{1}{- 3} d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 5

Simplify.

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Step 5.1

Move the negative in front of the fraction.

$2 \int_{9}^{- 12} e^{u} (- \frac{1}{3}) d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 5.2

Combine $e^{u}$ and $\frac{1}{3}$ .

$2 \int_{9}^{- 12} - \frac{e^{u}}{3} d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 6

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

$2 (- \int_{9}^{- 12} \frac{e^{u}}{3} d u) + \int_{- 3}^{4} - 3 e^{x} d x$

Step 7

Multiply $- 1$ by $2$ .

$- 2 \int_{9}^{- 12} \frac{e^{u}}{3} d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 8

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- 2 (\frac{1}{3} \int_{9}^{- 12} e^{u} d u) + \int_{- 3}^{4} - 3 e^{x} d x$

Step 9

Simplify.

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Step 9.1

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

$\frac{- 2}{3} \int_{9}^{- 12} e^{u} d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 9.2

Move the negative in front of the fraction.

$- \frac{2}{3} \int_{9}^{- 12} e^{u} d u + \int_{- 3}^{4} - 3 e^{x} d x$

Step 10

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

$- \frac{2}{3} e^{u}]_{9}^{- 12} + \int_{- 3}^{4} - 3 e^{x} d x$

Step 11

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- \frac{2}{3} e^{u}]_{9}^{- 12} - 3 \int_{- 3}^{4} e^{x} d x$

Step 12

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

$- \frac{2}{3} e^{u}]_{9}^{- 12} - 3 (e^{x}]_{- 3}^{4})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $e^{u}$ at $- 12$ and at $9$ .

$- \frac{2}{3} ((e^{- 12}) - e^{9}) - 3 (e^{x}]_{- 3}^{4})$

Step 13.2

Evaluate $e^{x}$ at $4$ and at $- 3$ .

$- \frac{2}{3} ((e^{- 12}) - e^{9}) - 3 ((e^{4}) - e^{- 3})$

Step 13.3

Remove parentheses.

$- \frac{2}{3} (e^{- 12} - e^{9}) - 3 (e^{4} - e^{- 3})$

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

Apply the distributive property.

$- \frac{2}{3} e^{- 12} - \frac{2}{3} (- e^{9}) - 3 (e^{4} - e^{- 3})$

Step 14.1.2

Combine $e^{- 12}$ and $\frac{2}{3}$ .

$- \frac{e^{- 12} \cdot 2}{3} - \frac{2}{3} (- e^{9}) - 3 (e^{4} - e^{- 3})$

Step 14.1.3

Multiply $- \frac{2}{3} (- e^{9})$ .

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Step 14.1.3.1

Multiply $- 1$ by $- 1$ .

$- \frac{e^{- 12} \cdot 2}{3} + 1 (\frac{2}{3}) e^{9} - 3 (e^{4} - e^{- 3})$

Step 14.1.3.2

Multiply $\frac{2}{3}$ by $1$ .

$- \frac{e^{- 12} \cdot 2}{3} + \frac{2}{3} e^{9} - 3 (e^{4} - e^{- 3})$

Step 14.1.3.3

Combine $\frac{2}{3}$ and $e^{9}$ .

$- \frac{e^{- 12} \cdot 2}{3} + \frac{2 e^{9}}{3} - 3 (e^{4} - e^{- 3})$

Step 14.1.4

Move $e^{- 12}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 (e^{4} - e^{- 3})$

Step 14.1.5

Apply the distributive property.

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 e^{4} - 3 (- e^{- 3})$

Step 14.1.6

Multiply $- 1$ by $- 3$ .

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 e^{4} + 3 e^{- 3}$

Step 14.2

Simplify each term.

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Step 14.2.1

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

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 e^{4} + 3 \frac{1}{e^{3}}$

Step 14.2.2

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

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 e^{4} + \frac{3}{e^{3}}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3 e^{12}} + \frac{2 e^{9}}{3} - 3 e^{4} + \frac{3}{e^{3}}$

Decimal Form:

$5238.41085872 \dots$

Step 16