Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

$- x^{- 2}$

Step 1.1.4

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

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

Step 1.2

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

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

Step 1.3

Divide $1$ by $1$ .

$u_{lower} = 1$

Step 1.4

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

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

Step 1.5

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

$u_{lower} = 1$

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

Step 1.6

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

$\int_{1}^{\frac{1}{2}} - e^{u} d u$

Step 2

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

$- \int_{1}^{\frac{1}{2}} e^{u} d u$

Step 3

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

$- (e^{u}]_{1}^{\frac{1}{2}})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $e^{u}$ at $\frac{1}{2}$ and at $1$ .

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

Step 4.2

Simplify.

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- e^{\frac{1}{2}} - - e$

Step 5.2

Multiply $- - e$ .

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2

Multiply $e$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.06956055 \dots$

Step 7