Calculus Examples

$\int_{- 8}^{0} x e^{- 4 x} d x$

Step 1

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

$x (- \frac{1}{4} e^{- 4 x})]_{- 8}^{0} - \int_{- 8}^{0} - \frac{1}{4} e^{- 4 x} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$x (- \frac{e^{- 4 x}}{4})]_{- 8}^{0} - \int_{- 8}^{0} - \frac{1}{4} e^{- 4 x} d x$

Step 2.2

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} - \int_{- 8}^{0} - \frac{1}{4} e^{- 4 x} d x$

Step 3

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} - (- \frac{1}{4} \int_{- 8}^{0} e^{- 4 x} d x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + 1 (\frac{1}{4} \int_{- 8}^{0} e^{- 4 x} d x)$

Step 4.2

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} \int_{- 8}^{0} e^{- 4 x} d x$

Step 5

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

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Differentiate $- 4 x$ .

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

Step 5.1.2

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 5.1.3

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

$- 4 \cdot 1$

Step 5.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 5.2

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

$u_{lower} = - 4 \cdot - 8$

Step 5.3

Multiply $- 4$ by $- 8$ .

$u_{lower} = 32$

Step 5.4

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

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

Step 5.5

Multiply $- 4$ by $0$ .

$u_{upper} = 0$

Step 5.6

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

$u_{lower} = 32$

$u_{upper} = 0$

Step 5.7

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} \int_{32}^{0} e^{u} \frac{1}{- 4} d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Move the negative in front of the fraction.

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} \int_{32}^{0} e^{u} (- \frac{1}{4}) d u$

Step 6.2

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} \int_{32}^{0} - \frac{e^{u}}{4} d u$

Step 7

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} (- \int_{32}^{0} \frac{e^{u}}{4} d u)$

Step 8

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} + \frac{1}{4} (- (\frac{1}{4} \int_{32}^{0} e^{u} d u))$

Step 9

Simplify.

Tap for more steps...

Step 9.1

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} - \frac{1}{4 \cdot 4} \int_{32}^{0} e^{u} d u$

Step 9.2

Multiply $4$ by $4$ .

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} - \frac{1}{16} \int_{32}^{0} e^{u} d u$

Step 10

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

$- \frac{x e^{- 4 x}}{4}]_{- 8}^{0} - \frac{1}{16} e^{u}]_{32}^{0}$

Step 11

Substitute and simplify.

Tap for more steps...

Step 11.1

Evaluate $- \frac{x e^{- 4 x}}{4}$ at $0$ and at $- 8$ .

$(- \frac{0 e^{- 4 \cdot 0}}{4}) + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} e^{u}]_{32}^{0}$

Step 11.2

Evaluate $e^{u}$ at $0$ and at $32$ .

$(- \frac{0 e^{- 4 \cdot 0}}{4}) + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Multiply $- 4$ by $0$ .

$- \frac{0 e^{0}}{4} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.2

Anything raised to $0$ is $1$ .

$- \frac{0 \cdot 1}{4} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.3

Multiply $0$ by $1$ .

$- \frac{0}{4} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.4

Cancel the common factor of $0$ and $4$ .

Tap for more steps...

Step 11.3.4.1

Factor $4$ out of $0$ .

$- \frac{4 (0)}{4} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.4.2

Cancel the common factors.

Tap for more steps...

Step 11.3.4.2.1

Factor $4$ out of $4$ .

$- \frac{4 \cdot 0}{4 \cdot 1} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.4.2.2

Cancel the common factor.

$- \frac{4 \cdot 0}{4 \cdot 1} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.4.2.3

Rewrite the expression.

$- \frac{0}{1} + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.4.2.4

Divide $0$ by $1$ .

$- 0 + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.5

Multiply $- 1$ by $0$ .

$0 + \frac{- 8 e^{- 4 \cdot - 8}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.6

Multiply $- 4$ by $- 8$ .

$0 + \frac{- 8 e^{32}}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.7

Cancel the common factor of $- 8$ and $4$ .

Tap for more steps...

Step 11.3.7.1

Factor $4$ out of $- 8 e^{32}$ .

$0 + \frac{4 (- 2 e^{32})}{4} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.7.2

Cancel the common factors.

Tap for more steps...

Step 11.3.7.2.1

Factor $4$ out of $4$ .

$0 + \frac{4 (- 2 e^{32})}{4 (1)} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.7.2.2

Cancel the common factor.

$0 + \frac{4 (- 2 e^{32})}{4 \cdot 1} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.7.2.3

Rewrite the expression.

$0 + \frac{- 2 e^{32}}{1} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.7.2.4

Divide $- 2 e^{32}$ by $1$ .

$0 - 2 e^{32} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.8

Subtract $2 e^{32}$ from $0$ .

$- 2 e^{32} - \frac{1}{16} ((e^{0}) - e^{32})$

Step 11.3.9

Anything raised to $0$ is $1$ .

$- 2 e^{32} - \frac{1}{16} (1 - e^{32})$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Apply the distributive property.

$- 2 e^{32} - \frac{1}{16} \cdot 1 - \frac{1}{16} (- e^{32})$

Step 12.1.2

Multiply $- 1$ by $1$ .

$- 2 e^{32} - \frac{1}{16} - \frac{1}{16} (- e^{32})$

Step 12.1.3

Multiply $- \frac{1}{16} (- e^{32})$ .

Tap for more steps...

Step 12.1.3.1

Multiply $- 1$ by $- 1$ .

$- 2 e^{32} - \frac{1}{16} + 1 (\frac{1}{16}) e^{32}$

Step 12.1.3.2

Multiply $\frac{1}{16}$ by $1$ .

$- 2 e^{32} - \frac{1}{16} + \frac{1}{16} e^{32}$

Step 12.1.3.3

Combine $\frac{1}{16}$ and $e^{32}$ .

$- 2 e^{32} - \frac{1}{16} + \frac{e^{32}}{16}$

Step 12.2

To write $- 2 e^{32}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- 2 e^{32} \cdot \frac{16}{16} + \frac{e^{32}}{16} - \frac{1}{16}$

Step 12.3

Combine $- 2 e^{32}$ and $\frac{16}{16}$ .

$\frac{- 2 e^{32} \cdot 16}{16} + \frac{e^{32}}{16} - \frac{1}{16}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{- 2 e^{32} \cdot 16 + e^{32}}{16} - \frac{1}{16}$

Step 12.5

Combine the numerators over the common denominator.

$\frac{- 2 e^{32} \cdot 16 + e^{32} - 1}{16}$

Step 12.6

Multiply $16$ by $- 2$ .

$\frac{- 32 e^{32} + e^{32} - 1}{16}$

Step 12.7

Add $- 32 e^{32}$ and $e^{32}$ .

$\frac{- 31 e^{32} - 1}{16}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{- 31 e^{32} - 1}{16}$

Decimal Form:

$- 152990735353944$

Step 14