Calculus Examples

$\int \frac{- e^{\frac{1}{x}}}{x^{2}} d x$

Step 1

Move the negative in front of the fraction.

$\int - \frac{e^{\frac{1}{x}}}{x^{2}} d x$

Step 2

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

$- \int \frac{e^{\frac{1}{x}}}{x^{2}} d x$

Step 3

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 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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 3.1.4

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

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

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$- \int - e^{u} d u$

Step 4

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

$- - \int e^{u} d u$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$1 \int e^{u} d u$

Step 5.2

Multiply $\int e^{u} d u$ by $1$ .

$\int e^{u} d u$

Step 6

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

$e^{u} + C$

Step 7

Replace all occurrences of $u$ with $\frac{1}{x}$ .

$e^{\frac{1}{x}} + C$