Calculus Examples

$\int_{0}^{1} e^{- 4 x} d x$

Step 1

Let $u = - 4 x$ . Then $d u = - 4 d x$ , so $- \frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = - 4 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 1.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 1.1.3

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

$- 4 \cdot 1$

Step 1.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 1.2

Substitute the lower limit in for $x$ in $u = - 4 x$ .

$u_{lower} = - 4 \cdot 0$

Step 1.3

Multiply $- 4$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = - 4 x$ .

$u_{upper} = - 4 \cdot 1$

Step 1.5

Multiply $- 4$ by $1$ .

$u_{upper} = - 4$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = - 4$

Step 1.7

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

$\int_{0}^{- 4} e^{u} \frac{1}{- 4} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{0}^{- 4} e^{u} (- \frac{1}{4}) d u$

Step 2.2

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

$\int_{0}^{- 4} - \frac{e^{u}}{4} d u$

Step 3

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

$- \int_{0}^{- 4} \frac{e^{u}}{4} d u$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- (\frac{1}{4} \int_{0}^{- 4} e^{u} d u)$

Step 5

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

$- \frac{1}{4} e^{u}]_{0}^{- 4}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $e^{u}$ at $- 4$ and at $0$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Anything raised to $0$ is $1$ .

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

Step 6.2.2

Multiply $- 1$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$- \frac{1}{4} e^{- 4} - \frac{1}{4} \cdot - 1$

Step 7.2

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

$- \frac{e^{- 4}}{4} - \frac{1}{4} \cdot - 1$

Step 7.3

Multiply $- \frac{1}{4} \cdot - 1$ .

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.3.2

Multiply $\frac{1}{4}$ by $1$ .

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

Step 7.4

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.24542109 \dots$

Step 9