Calculus Examples

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

Step 1

Let $u = \frac{1}{x^{5}}$ . Then $d u = - \frac{5}{x^{6}} d x$ , so $1 d u = - \frac{5}{x^{6}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Apply basic rules of exponents.

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Step 1.1.2.1

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

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

Step 1.1.2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 1.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.2.2.2

Multiply $5$ by $- 1$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 5$ .

$- 5 x^{- 6}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 5$ and $\frac{1}{x^{6}}$ .

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

Step 1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{1}{x^{5}}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

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

Step 1.3.2

Divide $1$ by $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{1}{x^{5}}$ .

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

Step 1.5

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

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

Step 1.6

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

$u_{lower} = 1$

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

Step 1.7

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

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

Step 2

Since $- \frac{1}{5}$ is constant with respect to $u$ , move $- \frac{1}{5}$ out of the integral.

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $e^{u}$ at $\frac{1}{32}$ and at $1$ .

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

Step 4.2

Simplify.

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine $e^{\frac{1}{32}}$ and $\frac{1}{5}$ .

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

Step 5.3

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

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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