Calculus Examples

$\int_{0}^{5} 3000 e^{- 0.05 x} d x$

Step 1

Since $3000$ is constant with respect to $x$ , move $3000$ out of the integral.

$3000 \int_{0}^{5} e^{- 0.05 x} d x$

Step 2

Let $u = - 0.05 x$ . Then $d u = - 0.05 d x$ , so $\frac{1}{- 0.05} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = - 0.05 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $- 0.05 x$ .

$\frac{d}{d x} [- 0.05 x]$

Step 2.1.2

Since $- 0.05$ is constant with respect to $x$ , the derivative of $- 0.05 x$ with respect to $x$ is $- 0.05 \frac{d}{d x} [x]$ .

$- 0.05 \frac{d}{d x} [x]$

Step 2.1.3

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

$- 0.05 \cdot 1$

Step 2.1.4

Multiply $- 0.05$ by $1$ .

$- 0.05$

Step 2.2

Substitute the lower limit in for $x$ in $u = - 0.05 x$ .

$u_{lower} = - 0.05 \cdot 0$

Step 2.3

Multiply $- 0.05$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = - 0.05 x$ .

$u_{upper} = - 0.05 \cdot 5$

Step 2.5

Multiply $- 0.05$ by $5$ .

$u_{upper} = - 0.25$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = - 0.25$

Step 2.7

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

$3000 \int_{0}^{- 0.25} e^{u} \frac{1}{- 0.05} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$3000 \int_{0}^{- 0.25} e^{u} (- \frac{1}{0.05}) d u$

Step 3.2

Combine $e^{u}$ and $\frac{1}{0.05}$ .

$3000 \int_{0}^{- 0.25} - \frac{e^{u}}{0.05} d u$

Step 4

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

$3000 (- \int_{0}^{- 0.25} \frac{e^{u}}{0.05} d u)$

Step 5

Multiply $- 1$ by $3000$ .

$- 3000 \int_{0}^{- 0.25} \frac{e^{u}}{0.05} d u$

Step 6

Since $\frac{1}{0.05}$ is constant with respect to $u$ , move $\frac{1}{0.05}$ out of the integral.

$- 3000 (\frac{1}{0.05} \int_{0}^{- 0.25} e^{u} d u)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{0.05}$ and $- 3000$ .

$\frac{- 3000}{0.05} \int_{0}^{- 0.25} e^{u} d u$

Step 7.2

Move the negative in front of the fraction.

$- \frac{3000}{0.05} \int_{0}^{- 0.25} e^{u} d u$

Step 8

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

$- \frac{3000}{0.05} e^{u}]_{0}^{- 0.25}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $e^{u}$ at $- 0.25$ and at $0$ .

$- \frac{3000}{0.05} ((e^{- 0.25}) - e^{0})$

Step 9.2

Simplify.

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Step 9.2.1

Anything raised to $0$ is $1$ .

$- \frac{3000}{0.05} (e^{- 0.25} - 1 \cdot 1)$

Step 9.2.2

Multiply $- 1$ by $1$ .

$- \frac{3000}{0.05} (e^{- 0.25} - 1)$

Step 10

Simplify.

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Step 10.1

Divide $3000$ by $0.05$ .

$- 1 \cdot 60000 (e^{- 0.25} - 1)$

Step 10.2

Multiply $- 1$ by $60000$ .

$- 60000 (e^{- 0.25} - 1)$

Step 10.3

Apply the distributive property.

$- 60000 e^{- 0.25} - 60000 \cdot - 1$

Step 10.4

Multiply $- 60000$ by $- 1$ .

$- 60000 e^{- 0.25} + 60000$

Step 10.5

Simplify each term.

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Step 10.5.1

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

$- 60000 \frac{1}{e^{0.25}} + 60000$

Step 10.5.2

Combine $- 60000$ and $\frac{1}{e^{0.25}}$ .

$\frac{- 60000}{e^{0.25}} + 60000$

Step 10.5.3

Move the negative in front of the fraction.

$- \frac{60000}{e^{0.25}} + 60000$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{60000}{e^{0.25}} + 60000$

Decimal Form:

$13271.95301571 \dots$

Step 12