Calculus Examples

$\int_{1}^{3} \frac{e^{\frac{3}{x}}}{x^{2}} d x$

Step 1

Let $u = \frac{3}{x}$ . Then $d u = - \frac{3}{x^{2}} d x$ , so $1 d u = - \frac{3}{x^{2}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{3}{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{3}{x}$ .

$\frac{d}{d x} [\frac{3}{x}]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

$3 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.3

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

$3 \frac{d}{d x} [x^{- 1}]$

Step 1.1.4

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

$3 (- x^{- 2})$

Step 1.1.5

Multiply $- 1$ by $3$ .

$- 3 x^{- 2}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

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

$- 3 \frac{1}{x^{2}}$

Step 1.1.6.2

Combine terms.

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Step 1.1.6.2.1

Combine $- 3$ and $\frac{1}{x^{2}}$ .

$\frac{- 3}{x^{2}}$

Step 1.1.6.2.2

Move the negative in front of the fraction.

$- \frac{3}{x^{2}}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{3}{x}$ .

$u_{lower} = \frac{3}{1}$

Step 1.3

Divide $3$ by $1$ .

$u_{lower} = 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{3}{x}$ .

$u_{upper} = \frac{3}{3}$

Step 1.5

Divide $3$ by $3$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 3$

$u_{upper} = 1$

Step 1.7

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

$\int_{3}^{1} - \frac{1}{3} e^{u} d u$

Step 2

Since $- \frac{1}{3}$ is constant with respect to $u$ , move $- \frac{1}{3}$ out of the integral.

$- \frac{1}{3} \int_{3}^{1} e^{u} d u$

Step 3

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

$- \frac{1}{3} e^{u}]_{3}^{1}$

Step 4

Evaluate $e^{u}$ at $1$ and at $3$ .

$- \frac{1}{3} ((e^{1}) - e^{3})$

Step 5

Simplify.

$- \frac{1}{3} (e - e^{3})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3} \cdot (e - e^{3})$

Decimal Form:

$5.78908503 \dots$