Calculus Examples

\int_{1}^{2} \frac{e^{\frac{1}{x^{5}}}}{x^{6}} d x

$\int_{1}^{2} \frac{e^{\frac{1}{x^{5}}}}{x^{6}} d x$

Step 1

Let

u = \frac{1}{x^{5}}

$u = \frac{1}{x^{5}}$ . Then

d u = - \frac{5}{x^{6}} d x

$d u = - \frac{5}{x^{6}} d x$ , so

1 d u = - \frac{5}{x^{6}} d x

$1 d u = - \frac{5}{x^{6}} d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = \frac{1}{x^{5}}

$u = \frac{1}{x^{5}}$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

\frac{1}{x^{5}}

$\frac{1}{x^{5}}$ .

\frac{d}{d x} [\frac{1}{x^{5}}]

$\frac{d}{d x} [\frac{1}{x^{5}}]$

Step 1.1.2

Apply basic rules of exponents.

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Step 1.1.2.1

Rewrite

\frac{1}{x^{5}}

$\frac{1}{x^{5}}$ as

{(x^{5})}^{- 1}

${(x^{5})}^{- 1}$ .

\frac{d}{d x} [{(x^{5})}^{- 1}]

$\frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 1.1.2.2

Multiply the exponents in

{(x^{5})}^{- 1}

${(x^{5})}^{- 1}$ .

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Step 1.1.2.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{d}{d x} [x^{5 \cdot - 1}]

$\frac{d}{d x} [x^{5 \cdot - 1}]$

Step 1.1.2.2.2

Multiply

5

$5$ by

- 1

$- 1$ .

\frac{d}{d x} [x^{- 5}]

$\frac{d}{d x} [x^{- 5}]$

\frac{d}{d x} [x^{- 5}]

$\frac{d}{d x} [x^{- 5}]$

\frac{d}{d x} [x^{- 5}]

$\frac{d}{d x} [x^{- 5}]$

Step 1.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = - 5

$n = - 5$ .

- 5 x^{- 6}

$- 5 x^{- 6}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

- 5 \frac{1}{x^{6}}

$- 5 \frac{1}{x^{6}}$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine

- 5

$- 5$ and

\frac{1}{x^{6}}

$\frac{1}{x^{6}}$ .

\frac{- 5}{x^{6}}

$\frac{- 5}{x^{6}}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

- \frac{5}{x^{6}}

$- \frac{5}{x^{6}}$

- \frac{5}{x^{6}}

$- \frac{5}{x^{6}}$

- \frac{5}{x^{6}}

$- \frac{5}{x^{6}}$

- \frac{5}{x^{6}}

$- \frac{5}{x^{6}}$

Step 1.2

Substitute the lower limit in for

x

$x$ in

u = \frac{1}{x^{5}}

$u = \frac{1}{x^{5}}$ .

u_{lower} = \frac{1}{1^{5}}

$u_{lower} = \frac{1}{1^{5}}$

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

u_{lower} = \frac{1}{1}

$u_{lower} = \frac{1}{1}$

Step 1.3.2

Divide

1

$1$ by

1

$1$ .

u_{lower} = 1

$u_{lower} = 1$

u_{lower} = 1

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for

x

$x$ in

u = \frac{1}{x^{5}}

$u = \frac{1}{x^{5}}$ .

u_{upper} = \frac{1}{2^{5}}

$u_{upper} = \frac{1}{2^{5}}$

Step 1.5

Raise

2

$2$ to the power of

5

$5$ .

u_{upper} = \frac{1}{32}

$u_{upper} = \frac{1}{32}$

Step 1.6

The values found for

u_{lower}

$u_{lower}$ and

u_{upper}

$u_{upper}$ will be used to evaluate the definite integral.

u_{lower} = 1

$u_{lower} = 1$

u_{upper} = \frac{1}{32}

$u_{upper} = \frac{1}{32}$

Step 1.7

Rewrite the problem using

u

$u$ ,

d u

$d u$ , and the new limits of integration.

\int_{1}^{\frac{1}{32}} - \frac{1}{5} e^{u} d u

$\int_{1}^{\frac{1}{32}} - \frac{1}{5} e^{u} d u$

\int_{1}^{\frac{1}{32}} - \frac{1}{5} e^{u} d u

$\int_{1}^{\frac{1}{32}} - \frac{1}{5} e^{u} d u$

Step 2

Since

- \frac{1}{5}

$- \frac{1}{5}$ is constant with respect to

u

$u$ , move

- \frac{1}{5}

$- \frac{1}{5}$ out of the integral.

- \frac{1}{5} \int_{1}^{\frac{1}{32}} e^{u} d u

$- \frac{1}{5} \int_{1}^{\frac{1}{32}} e^{u} d u$

Step 3

The integral of

e^{u}

$e^{u}$ with respect to

u

$u$ is

e^{u}

$e^{u}$ .

- \frac{1}{5} e^{u}]_{1}^{\frac{1}{32}}

$- \frac{1}{5} e^{u}]_{1}^{\frac{1}{32}}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate

e^{u}

$e^{u}$ at

\frac{1}{32}

$\frac{1}{32}$ and at

1

$1$ .

- \frac{1}{5} ((e^{\frac{1}{32}}) - e^{1})

$- \frac{1}{5} ((e^{\frac{1}{32}}) - e^{1})$

Step 4.2

Simplify.

- \frac{1}{5} (e^{\frac{1}{32}} - e)

$- \frac{1}{5} (e^{\frac{1}{32}} - e)$

- \frac{1}{5} (e^{\frac{1}{32}} - e)

$- \frac{1}{5} (e^{\frac{1}{32}} - e)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

- \frac{1}{5} e^{\frac{1}{32}} - \frac{1}{5} (- e)

$- \frac{1}{5} e^{\frac{1}{32}} - \frac{1}{5} (- e)$

Step 5.2

Combine

e^{\frac{1}{32}}

$e^{\frac{1}{32}}$ and

\frac{1}{5}

$\frac{1}{5}$ .

- \frac{e^{\frac{1}{32}}}{5} - \frac{1}{5} (- e)

$- \frac{e^{\frac{1}{32}}}{5} - \frac{1}{5} (- e)$

Step 5.3

Multiply

- \frac{1}{5} (- e)

$- \frac{1}{5} (- e)$ .

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Step 5.3.1

Multiply

$- 1$ by

$- 1$ .

$- \frac{e^{\frac{1}{32}}}{5} + 1 (\frac{1}{5}) e$

Step 5.3.2

Multiply

$\frac{1}{5}$ by

$1$ .

$- \frac{e^{\frac{1}{32}}}{5} + \frac{1}{5} e$

Step 5.3.3

Combine

$\frac{1}{5}$ and

$e$ .

$- \frac{e^{\frac{1}{32}}}{5} + \frac{e}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{e^{\frac{1}{32}}}{5} + \frac{e}{5}$

Decimal Form:

$0.33730768 \dots$

Step 7