Calculus Examples

$\int_{0}^{8} 8 e^{- x} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{8} e^{- x} d x$

Step 2

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $- x$ .

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

Step 2.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 2.1.3

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

$- 1 \cdot 1$

Step 2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.2

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

$u_{lower} = - 0$

Step 2.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = - 1 \cdot 8$

Step 2.5

Multiply $- 1$ by $8$ .

$u_{upper} = - 8$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = - 8$

Step 2.7

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

$8 \int_{0}^{- 8} - e^{u} d u$

Step 3

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

$8 (- \int_{0}^{- 8} e^{u} d u)$

Step 4

Multiply $- 1$ by $8$ .

$- 8 \int_{0}^{- 8} e^{u} d u$

Step 5

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

$- 8 (e^{u}]_{0}^{- 8})$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $e^{u}$ at $- 8$ and at $0$ .

$- 8 ((e^{- 8}) - e^{0})$

Step 6.2

Simplify.

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Step 6.2.1

Anything raised to $0$ is $1$ .

$- 8 (e^{- 8} - 1 \cdot 1)$

Step 6.2.2

Multiply $- 1$ by $1$ .

$- 8 (e^{- 8} - 1)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$- 8 e^{- 8} - 8 \cdot - 1$

Step 7.2

Multiply $- 8$ by $- 1$ .

$- 8 e^{- 8} + 8$

Step 7.3

Simplify each term.

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Step 7.3.1

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

$- 8 \frac{1}{e^{8}} + 8$

Step 7.3.2

Combine $- 8$ and $\frac{1}{e^{8}}$ .

$\frac{- 8}{e^{8}} + 8$

Step 7.3.3

Move the negative in front of the fraction.

$- \frac{8}{e^{8}} + 8$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{e^{8}} + 8$

Decimal Form:

$7.99731629 \dots$

Step 9