Calculus Examples

$\int_{0}^{2} (4 x + 5) e^{- x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 4 x + 5$ and $d v = e^{- x}$ .

$(4 x + 5) (- e^{- x})]_{0}^{2} - \int_{0}^{2} - e^{- x} \cdot 4 d x$

Step 2

Multiply $4$ by $- 1$ .

$(4 x + 5) (- e^{- x})]_{0}^{2} - \int_{0}^{2} - 4 e^{- x} d x$

Step 3

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

$(4 x + 5) (- e^{- x})]_{0}^{2} - (- 4 \int_{0}^{2} e^{- x} d x)$

Step 4

Multiply $- 4$ by $- 1$ .

$(4 x + 5) (- e^{- x})]_{0}^{2} + 4 \int_{0}^{2} e^{- x} d x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $- x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 5.2

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

$u_{lower} = - 0$

Step 5.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 5.4

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

$u_{upper} = - 1 \cdot 2$

Step 5.5

Multiply $- 1$ by $2$ .

$u_{upper} = - 2$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = - 2$

Step 5.7

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

$(4 x + 5) (- e^{- x})]_{0}^{2} + 4 \int_{0}^{- 2} - e^{u} d u$

Step 6

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

$(4 x + 5) (- e^{- x})]_{0}^{2} + 4 (- \int_{0}^{- 2} e^{u} d u)$

Step 7

Multiply $- 1$ by $4$ .

$(4 x + 5) (- e^{- x})]_{0}^{2} - 4 \int_{0}^{- 2} e^{u} d u$

Step 8

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

$(4 x + 5) (- e^{- x})]_{0}^{2} - 4 (e^{u}]_{0}^{- 2})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $(4 x + 5) (- e^{- x})$ at $2$ and at $0$ .

$((4 \cdot 2 + 5) (- e^{- 1 \cdot 2})) - (4 \cdot 0 + 5) (- e^{- 0}) - 4 (e^{u}]_{0}^{- 2})$

Step 9.2

Evaluate $e^{u}$ at $- 2$ and at $0$ .

$((4 \cdot 2 + 5) (- e^{- 1 \cdot 2})) - (4 \cdot 0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $4$ by $2$ .

$(8 + 5) (- e^{- 1 \cdot 2}) - (4 \cdot 0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.2

Add $8$ and $5$ .

$13 (- e^{- 1 \cdot 2}) - (4 \cdot 0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.3

Multiply $- 1$ by $2$ .

$13 (- e^{- 2}) - (4 \cdot 0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.4

Multiply $- 1$ by $13$ .

$- 13 e^{- 2} - (4 \cdot 0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.5

Multiply $4$ by $0$ .

$- 13 e^{- 2} - (0 + 5) (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.6

Add $0$ and $5$ .

$- 13 e^{- 2} - 1 \cdot 5 (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.7

Multiply $- 1$ by $5$ .

$- 13 e^{- 2} - 5 (- e^{- 0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.8

Multiply $- 1$ by $0$ .

$- 13 e^{- 2} - 5 (- e^{0}) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.9

Anything raised to $0$ is $1$ .

$- 13 e^{- 2} - 5 (- 1 \cdot 1) - 4 ((e^{- 2}) - e^{0})$

Step 9.3.10

Multiply $- 1$ by $1$ .

$- 13 e^{- 2} - 5 \cdot - 1 - 4 ((e^{- 2}) - e^{0})$

Step 9.3.11

Multiply $- 5$ by $- 1$ .

$- 13 e^{- 2} + 5 - 4 ((e^{- 2}) - e^{0})$

Step 9.3.12

Anything raised to $0$ is $1$ .

$- 13 e^{- 2} + 5 - 4 (e^{- 2} - 1 \cdot 1)$

Step 9.3.13

Multiply $- 1$ by $1$ .

$- 13 e^{- 2} + 5 - 4 (e^{- 2} - 1)$

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

$- 13 \frac{1}{e^{2}} + 5 - 4 (e^{- 2} - 1)$

Step 10.1.2

Combine $- 13$ and $\frac{1}{e^{2}}$ .

$\frac{- 13}{e^{2}} + 5 - 4 (e^{- 2} - 1)$

Step 10.1.3

Move the negative in front of the fraction.

$- \frac{13}{e^{2}} + 5 - 4 (e^{- 2} - 1)$

Step 10.1.4

Apply the distributive property.

$- \frac{13}{e^{2}} + 5 - 4 e^{- 2} - 4 \cdot - 1$

Step 10.1.5

Multiply $- 4$ by $- 1$ .

$- \frac{13}{e^{2}} + 5 - 4 e^{- 2} + 4$

Step 10.2

Simplify each term.

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Step 10.2.1

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

$- \frac{13}{e^{2}} + 5 - 4 \frac{1}{e^{2}} + 4$

Step 10.2.2

Combine $- 4$ and $\frac{1}{e^{2}}$ .

$- \frac{13}{e^{2}} + 5 + \frac{- 4}{e^{2}} + 4$

Step 10.2.3

Move the negative in front of the fraction.

$- \frac{13}{e^{2}} + 5 - \frac{4}{e^{2}} + 4$

Step 10.3

Combine the numerators over the common denominator.

$5 + 4 + \frac{- 13 - 4}{e^{2}}$

Step 10.4

Subtract $4$ from $- 13$ .

$5 + 4 + \frac{- 17}{e^{2}}$

Step 10.5

Move the negative in front of the fraction.

$5 + 4 - \frac{17}{e^{2}}$

Step 10.6

Add $5$ and $4$ .

$9 - \frac{17}{e^{2}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$9 - \frac{17}{e^{2}}$

Decimal Form:

$6.69930018 \dots$

Step 12