Calculus Examples

$\int_{- 5}^{1} (- 5 x + 4 e^{- x}) d x$

Step 1

Remove parentheses.

$\int_{- 5}^{1} - 5 x + 4 e^{- x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 5}^{1} - 5 x d x + \int_{- 5}^{1} 4 e^{- x} d x$

Step 3

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 5 \int_{- 5}^{1} x d x + \int_{- 5}^{1} 4 e^{- x} d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 5 (\frac{1}{2} x^{2}]_{- 5}^{1}) + \int_{- 5}^{1} 4 e^{- x} d x$

Step 5

Combine $\frac{1}{2}$ and $x^{2}$ .

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) + \int_{- 5}^{1} 4 e^{- x} d x$

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) + 4 \int_{- 5}^{1} e^{- x} d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $- x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.2

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

$u_{lower} = - - 5$

Step 7.3

Multiply $- 1$ by $- 5$ .

$u_{lower} = 5$

Step 7.4

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

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

Step 7.5

Multiply $- 1$ by $1$ .

$u_{upper} = - 1$

Step 7.6

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

$u_{lower} = 5$

$u_{upper} = - 1$

Step 7.7

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

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) + 4 \int_{5}^{- 1} - e^{u} d u$

Step 8

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

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) + 4 (- \int_{5}^{- 1} e^{u} d u)$

Step 9

Multiply $- 1$ by $4$ .

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) - 4 \int_{5}^{- 1} e^{u} d u$

Step 10

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

$- 5 (\frac{x^{2}}{2}]_{- 5}^{1}) - 4 (e^{u}]_{5}^{- 1})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $- 5$ .

$- 5 ((\frac{1^{2}}{2}) - \frac{{(- 5)}^{2}}{2}) - 4 (e^{u}]_{5}^{- 1})$

Step 11.2

Evaluate $e^{u}$ at $- 1$ and at $5$ .

$- 5 ((\frac{1^{2}}{2}) - \frac{{(- 5)}^{2}}{2}) - 4 ((e^{- 1}) - e^{5})$

Step 11.3

Simplify.

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Step 11.3.1

One to any power is one.

$- 5 (\frac{1}{2} - \frac{{(- 5)}^{2}}{2}) - 4 ((e^{- 1}) - e^{5})$

Step 11.3.2

Raise $- 5$ to the power of $2$ .

$- 5 (\frac{1}{2} - \frac{25}{2}) - 4 ((e^{- 1}) - e^{5})$

Step 11.3.3

Combine the numerators over the common denominator.

$- 5 \frac{1 - 25}{2} - 4 ((e^{- 1}) - e^{5})$

Step 11.3.4

Subtract $25$ from $1$ .

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

Step 11.3.5

Cancel the common factor of $- 24$ and $2$ .

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Step 11.3.5.1

Factor $2$ out of $- 24$ .

$- 5 \frac{2 \cdot - 12}{2} - 4 ((e^{- 1}) - e^{5})$

Step 11.3.5.2

Cancel the common factors.

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Step 11.3.5.2.1

Factor $2$ out of $2$ .

$- 5 \frac{2 \cdot - 12}{2 (1)} - 4 ((e^{- 1}) - e^{5})$

Step 11.3.5.2.2

Cancel the common factor.

$- 5 \frac{2 \cdot - 12}{2 \cdot 1} - 4 ((e^{- 1}) - e^{5})$

Step 11.3.5.2.3

Rewrite the expression.

$- 5 (\frac{- 12}{1}) - 4 ((e^{- 1}) - e^{5})$

Step 11.3.5.2.4

Divide $- 12$ by $1$ .

$- 5 \cdot - 12 - 4 ((e^{- 1}) - e^{5})$

Step 11.3.6

Multiply $- 5$ by $- 12$ .

$60 - 4 (e^{- 1} - e^{5})$

Step 12

Simplify each term.

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Step 12.1

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

$60 - 4 (\frac{1}{e} - e^{5})$

Step 12.2

Apply the distributive property.

$60 - 4 \frac{1}{e} - 4 (- e^{5})$

Step 12.3

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

$60 + \frac{- 4}{e} - 4 (- e^{5})$

Step 12.4

Multiply $- 1$ by $- 4$ .

$60 + \frac{- 4}{e} + 4 e^{5}$

Step 12.5

Move the negative in front of the fraction.

$60 - \frac{4}{e} + 4 e^{5}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$60 - \frac{4}{e} + 4 e^{5}$

Decimal Form:

$652.18111864 \dots$

Step 14